IN  MEMORIAM 
FLORIAN  CAJORl     | 


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ELEMENTS  OF  ALGEBRA: 


XMBRACINO   ALSO 


\)\ji/J^ 


THE  THEORY  AND  APPLICATION  OF  LOGARITHMS; 


TOGETHER    WITH 


AN    APPENDIX, 


CONTAINING 

INFINITE   8ERIK8,   THE   GENERAL  THEORY    OF*    EQUATIONS,   AND 

THE    MOST   APPROVED   METHODS  OF    RESOLVING 

THE    HIGHER    EQUATIONS. 


BY  REV.   DAVIS  W.   CLARK,  A.M., 

V 

FBINCIPAIi  or  AMBIIIA  SBKIIIART 


•   •      •    •  • 


NEW-YORK: 
Harper  dc  Brothers,  82  Cliff-Street. 

1843. 


^^  .^      ^-^'^ 


dffJL 


CAJORI 


Entered,  according  to  Act  of  Congress,  in  the  year  1843,  by 

Harper  &  Brothers, 
In  the  Clerk's  Office  of  the  Southern  District  of  New-York. 


PREFACE 


The  object  of  this  treatise  is  to  present  to  the  student  a  full 
and  systematic  course  of  practical  and  theoretical  elementary 
Algebra.  With  this  object  steadily  in  view,  the  author  has 
made  no  effort  for  the  display  of  mathematical  genius,  but  has 
assiduously  applied  himself  to  the  preparation  of  a  text- book 
in  the  science.  Believing  that  original  discoveries  are  not 
best  adapted  to  beginners,  he  has  satisfied  himself  with  the 
humble  vocation  of  collecting,  arranging,  and  illustrating  the 
ample  materials  already  provided.  But  it  is  due  to  himself 
to  say  that  these  materials  have  all  been  re-wrought,  and 
not  a  few  of  them  re-written  several  times.  It  has  been  a 
constant  endeavour  to  make  everything  explicit,  and  also  to 
exhibit  it  in  the  simplest  possible  form.  By  this  means,  the 
author  has  been  enabled  to  embrace,  within  a  comparatively 
small  compass,  a  more  comprehensive  view  of  the  science  than 
can  be  found  in  any  text-book  on  the  subject  now  in  use. 

Among  the  works  which  have  shared,  and  still  stare  most 
largely  in  the  patronage  of  the  public,  isolated  parts  or  sub- 
■Jects  are  treated  with  great  ability  and  clearness ;  but,  in  some 
instances,  these  works  are  remarkably  deficient,  so  far  as  con- 
cerns any  methodical  arrangement  of  the  subjects  introduced, 
while  also  other  subjects  of  great  importance  are  omitted  alto- 
gether. That  these  books  force  their  way  into  public  patron- 
age is  not  surprising,  when,  on  the  other  hand,  those  treatises 
which  are  systematic  in  the  arrangement  of  topics  are,  in 
general,  too  theoretical  and  abstract  for  the  convenience  or 
profit  of  the  beginner,  or,  indeed,  of  the  practical  algebraist. 


f\A  A  r>CLCl 


VI  PREFACir. 


In  collecting  his  materials,  the  author  has  consulted  the  most 
approved  writers  upon  the  subject.  It  would  be  difficult,  if 
not  impossible,  to  point  out  the  precise  amount  of  his  indebt- 
edness to  each ;  yet  he  does  not  hesitate  to  acknowledge  it, 
nor  has  any  desire  of  appearing  original  led  him  to  remodel 
these  materials.  Indeed,  this  has  been  done  only  when  it  was 
necessary  in  order  to  preserve  the  unity  of  the  work,  or  to 
render  the  subjects  more  explicit.  In  arranging  and  digesting 
these  materials,  however,  the  author  has  been  fettered  by  no 
adopted  system.  Whatever  seemed  most  appropriate  to  his 
general  object,  and  in  keeping  with  the  general  plan  of  his 
■work,  he  has  freely  made  use  of,  at  all  times  having  reference 
to  the  wants  of  our  schools,  and  endeavouring  to  meet  them. 
How  far  this  object  has  been  attained,  he  now  leaves  the 
reader  to  judge,  claiming  only  for  himself  that  it  is  a  well- 
meant  contribution  to  elementary  education  in  an  important 
branch  of  science. 

For  the  article  on  "  Roots  of  Numbers,"  as  well  as  for  other 
valuable  assistance  in  the  preparation  of  this  work,  he  is  in- 
debted to  the  Rev.  Joseph  Cummings,  A.B.,  lecturer  on  Nat- 
ural Science  in  the  Amenia  Seminary. 

In  the  present  work,  Algebra  has  not  been  regarded  merely 
as  an  introduction  to  the  higher  branches  of  mathematics,  but 
also  as  a  means  of  unfolding  more  clearly  the  principles  and 
theory  of  common  arithmetic.  This  is  an  important  consid- 
eration. A  great  portion  of  the  students  in  our  academies  and 
schools  do  not  pursue  the  mathematical  course  beyond  alge- 
bra. Such,  aside  from  the  mental  discipline  acquired  in  its 
study,  derive  their  chief  advantage  from  the  superior  under- 
standing it  gives  them  of  common  arithmetic  ;  and  we  speak 
only  the  common  sentiment  of  the  better-informed  school- 
teachers, when  we  say  that  few,  if  any,  are  properly  qualified 
to  teach  arithmetic  w^ithout  a  knowledge  of  algebra.  The  au- 
thor has  not,  however,  limited  himself  to  this  object,  and  be- 


PREFACE.  Til 

lieves  the  work  will  fully  answer  all  the  necessary  requisitions 
of  an  introduction  to  the  higher  branches  of  mathematics. 

The  Logic  of  Algebra  is  an  object  that  should  not  be  lost 
sight  of  in  the  study ;  and  in  order  that  the  student  may  be 
exercised  in  this,  every  important  principle  has  been  explained 
and  demonstrated.  But,  at  the  same  time,  the  explanations 
have  been  made  simple,  and  the  demonstrations  put  in  such  a 
form  (especially  in  the  first  part  of  the  work),  that  they  can 
be  easily  comprehended  by  those  unaccustomed  to  the  rigid 
demonstrations  of  analytical  algebra.  In  the  higher  depart- 
ments of  mathematics,  it  is  undoubtedly  desirable  that  the  for- 
mality in  stating  every  proposition,  and  the  course  of  demon- 
stration required  by  the  precise  rules  of  logic,  should  be  adhe- 
red to.  But  in  algebra  the  case  is  different.  The  mind  of  the 
student  must  become  gradually  habituated  to  the  more  ab- 
stract modes  of  thinking  and  precise  methods  of  reasoning ; 
and,  as  algebra  is  commenced  in  so  early  a  part  of  the  course, 
a  certain  degree  of  familiarity,  rather  than  formality,  in  the 
statement  of  propositions  and  in  their  proof,  becomes  not  only 
excusable,  but  even  necessary.  The  author,  however,  has 
studied  precision  in  the  statement  of  propositions,  and  endeav- 
oured to  make  his  reasoning  explicit.  In  this  way  has  he  en- 
deavoured to  make  the  theory  obvious  and  satisfactory. 

Believing  that  a  knowledge  of  the  general  principles  of 
algebra  can  be  perfected  and  permanently  secured  only  by 
frequent  and  rigid  application,  the  author  has  endeavoured, 
throughout  the  work,  to  blend  theory  and  practice.  For  this 
purpose,  a  careful  selection  of  problems  and  exercises  has  been 
made  from  the  most  approved  authors. 

In  the  ninth  section  the  author  has  given  a  clear  and  con- 
cise view  of  the  theory  of  Logarithms,  and  a  method  of  cal- 
culating common  logarithms,  or  those  in  general  use,  so  expli- 
cit, and  yet  so  simple,  that  the  student  well  versed  in  propor- 


Vlll  PREFACE. 

tion  and  progression  may  be  able  to  calculate  them  with  ease 
and  facility. 

As  the  last  three  sections  treat  upon  subjects  that  are  sel- 
dom called  into  use  by  the  merely  practical  algebraist,  and 
yet  subjects  that  are  indispensable  as  an  introduction  to  the 
higher  departments  of  mathematics,  they  have  been  thrown 
into  the  form  of  an  Appendix. 

This  work  was  commenced,  and  has  been-  carried  to  its 
completion,  amid  the  arduous  duties  incident  to  the  charge  of 
a  large  and  flourishing  seminary  of  learning.  Yet  labour  and 
care  have  been  bestowed  upon  every  part  of  it,  and  that,  too, 
while  the  author  was  daily  engaged  in  instructing  classes  in 
this  interesting  and  important  branch  of  study  ;  and  if,  under 
these  circumstances,  he  has  been  able  to  discover  the  wants 
of  the  student,  and  adapt  his  work  to  meet  those  wants,  he 
will  feel  amply  compensated  for  his  toil. 

D.  W.  Clark. 

Amenia  Seminaiy,  March,  1843. 


CONTENTS. 


SECTION  L 

Preliminary  Remarks  (^  1-11) 13 

Definitions  (12-18) U 

Axioms  (19) 15 

Algebraic  Notation  (20-75) 16 

SECTION  IL 

Addition  (^  76-61) 25 

Subtraction  (82-87) 29 

MultipUcation  (88-99) 31 

Diviaion  (100-116) 38 

SECTION  m. 

Algebraic  Fractions  (^  117-126) 47 

Discussion  of  Signs  (127-133) 48 

Reduction  of  Mixed  Numbers  (134) .50 

Reduction  of  Improper  Fractions  (135) 51 

Of  the  Greatest  Common  Divisor  (136-138) 52 

To  Reduce  a  Fraction  to  its  Lowest  Terms  (139) 58 

The  Least  Common  Multiple  (140) 60 

Reduction  of  Fractions  to  a  Common  Denominator  (141)       .       .        .        .61 

Least  Common  Denominator  (142) 63 

Addition  of  Fractions  (143-144) 64 

Subtraction  of  Fractions  (145-146) 66 

Multiplication  of  Fractions  (147-148) 67 

Division  of  Fractions  (149-150) 69 

SECTION  IV. 

Of  Equations  (^151-160) 72 

Equations  of  the  First  Degree  involving  only  one  Unknown  Quantity  (161-168)  74 

Problems  (169) 81 

Equations  of  the  First  Degree  involving  more  than  one  Unknown  Quantity 

(170-173) 90 

Elimir»ation  by  Comparison  (174-175) 91 

"          by  SubstituUon  (176-177) 92 

"          by  Addition  or  Subtraction  (178-179) 94 

Eqaations  involving  two  Unknown  Quantities  (180) 95 

B 


X  CONTENTS. 

Problems  requiring  two  Unknown  Quantities  (181) 97 

Equations  involving  three  or  more  Unknown  Quantities  (182-185)         .        .102 
Problems  requiring      "  "  "  «'  (186-187)        .        .  10" 

SECTION  V. 

Generalization  of  Algebraic  Problems  (§  188-195) 110 

Demonstration  of  Theorems  (196-206) 114 

Algebraic  Demonstration  of  certain  Properties  of  Numbers  (207-213)    .        .119 

Reduction  of  Formulas  relating  to  Simple  Interest  (214-217)  .        .        .123 

"  "  "  "   Compound  Interest  (218-219)    .        .        .124 

"  •'  '*  "  Fellowship  (220-225)         .        .        .        .126 

Discussion  of  Equations  of  the  First  Degree  (226-231) 129 

Theory  of  Negative  Quantities  (232-234) 131 

Explanation  of  Symbols.    Infinity  (235-237) 133 

"  "  Infinitesimals  (239-240) 134 

"  "  Indetermination  (241-244) 135 

Inequations  (245-250) 136 

SECTION  VI. 

Involution  and  Powers  (^  251-258)  .        . 140 

Binomial  Theorem  (259-272) 147 

Roots  of  Numbers  (273-288) 155 

Evolution  of  Algebraic  Quantities  (289-295) .176 

Calculus  of  Radicals  (296-314) -      .        .        .184 

SECTION  VII. 

Of  Equations  exceeding  the  First  Degree  (<^  315-318) 199 

Pure  Equations  (319) 200 

Problems  producmg  Pure  Equations  (321) 208 

Affected  Equations  of  the  Second  Degree  (322) 210 

First  Method  of  completing  the  Square  (327) 211 

Second  Method  of  completing  the  Square  (328-330) 213 

Particular  Cases  of  Affected  Quadratic  Equations  (331-337)  .        .        .        ,  214 

Equations 218 

Problems  producing  Affected  Quadratic  Equations 221 

Discussion  of  the  General  Equation  of  the  Second  Degree  (338-351)     .        .  224 

SECTION  VIII. 

Ratio  (^  352-358) .  234 

Proportion  (359-372) 239 

Arithmetical  Progression  (373-379) 246 

Geometrical  Progression  (380-390) 261 

SECTION  IX. 

Theory  of  Logarithms  (<5  391-400) 257 

Computation  of  Logarithmic  Tables  (401-402) 264 

Application  of  Logarithms  (403) 265 


CONTENTS.  XI 

Multiplication  and  Division  by  Logarithms  (40i) 2C0 

Involution  and  Evolution  by  Logarithms  (405) 207 

Exponential  Equations  (406-408) 208 

Geometrical  Series  (409) 271 

Compound  Interest  (410) 273 

APPENDIX. 

SECTION  X. 
Permutations,  Arrangements,  and  Combinations  (^  411-418)         .        .        .  277 
General  Demonstration  of  the  Binomial  Theorem  (419-431)   ....  282 

Continued  Fractions  (432) 288 

Infinite  Series  (433) 294 

Expansion  of  Infinite  Series  (434-435) 294 

Indeterminate  Coefficients  (430-437) 297 

Summation  of  Infinite  Series  (438) 300 

Recurring  Series  (439-444) 302 

Method  of  Differences  (445-450) 306 

Reversion  of  Series  (451) 312 

SECTION  XL 

General  Properties  of  Equations  (^  452-453)   . 314 

Composition  of  Equations  (454-45G) 318 

Transformation  of  Equations  (457-402) 320 

SECTION  XII. 

BK80LUTI0N  OP  THE   HIGHER   EQUATIONS. 

Cardan's  Method  of  Resolving  Cubic  Equations  (^  403^00)         .        .        .332 
Young's        "  "  "  "  (467-472)  .        .         .337 

Des  Cartes'  "  "  Equations  of  the  Fourth  Degree  (473-476)  .  343 

Newton's  Method  of  Approximation  (477-480) 345 

Resolution  of  Higher  Equations  by  Trial  and  Error  (481-483)       .        .        .347 

Young's  Method  of  Resolving  Higher  Equations  (484) 351 

Notes 355 


NOTE. 

The  ill  health  of  the  author  while  the  work  was  in  course  of  publication  has 
prevented  him  from  devoting  that  personal  attention  to  a  final  examination  of  the 
proofs  that  he  desired.  This  he  would  offer  as  an  apology  for  any  errors  that 
may  have  been  overlooked. 


ELEMENTS  OF  ALGEBRA. 


SECTION  I. 

Preliminary  Remarks. — Definitions. — jJxioms. — .Algebraic 
Method  of  J^otation. 

PHELIMINAEY    REMARKS. 

1.  The  object  of  Mathematical  Science  is  to  investigate 
the  relations  of  quantities  and  the  properties  of  numbers. 

2.  Quantity,  or  magnitude,  is  a  general  term,  embracing 
everything  which  admits  of  increase,  diminution,  and  meas- 
urement.* Thus,  a  given  weight  or  bulk,  a  sum  of  money, 
or  a  number  of  yards,  are  quantities. 

3.  The  measurement  of  quantity  is  accomplished  by  means 
of  an  assumed  unit  or  standard  of  measure.  This  unit  must 
be  of  the  same  kind  as  the  quantity.  Thus,  the  measuring 
unit  of  money  is  one  dollar ;  of  a  line  is  one  inch,  foot,  or 
mile,  &c. ;  of  area  is  one  square  inch,  foot,  or  acre,  &c. 

4.  Jfumbers  are  symbols  adopted  to  facilitate  the  investi- 
gation of  quantities.  They  represent  a  unit  or  an  assem- 
blao-e  of  units.  Thus,  35,  42,  and  64  are  numbers  j  but  $35, 
42  cwt.,  and  64  acres,  are  quantities. 

5.  Whole  numbers,  as  4,  6,  15,  30,  &c.,  are  called  integers. 
Broken  numbers,  as  i,  |,  -j^,  &c.,  are  called /ramo;w. 

6.  Any  number  which  can  be  divided  by  2  without  pro- 
ducing a  fraction,  is  called  an  even  number  ;  and  all  numbers 
which  cannot  be  divided  by  2  without  producing  a  fraction, 
are  called  odd  numbers. 

7.  Numbers  are  also  distinguished  into  composite  and  prime 

*  See  Note  A. 


14  ELEMENTS    OF    ALGEBRA.  [sECT.  I. 

numbers.  Any  number  which  can  be  produced  by  multi- 
plying two  or  more  numbers  together,  each  of  which  is 
greater  than  a  unit^  is  called  a  composite  number,  as  4,  6,  12, 
20,  &c.  Numbers  which  admit  of  no  exact  divisor  except 
themselves  and  unity,  are  called  prime  numbers,  as  1,  2,,  3, 
6,  7,  11,  (Src. 

8.  The  foundation  of  mathematical  reasoning  is  laid  in  Defi' 
nitions  and  Axioms.  The  absolute  certainty  of  its  conclu- 
sions results  no  less  from  the  exactness  of  mathematical 
definitions,  and  the  clearness  and  simplicity  of  its  first  prin- 
ciples, than  from  the  nature  of  the  subjects  about  which  it 
is  employed. 

9.  A  Definition,  when  applied  to  language,  is  a  brief  ex- 
planation of  what  is  meant  by  a  word  or  phrase.  When  ap- 
plied to  "a  thing,  it  is  an  analysis  of  its  parts  or  an  enumera- 
tion of  its  principal  attributes  ;  but  this  analysis  or  enumer- 
ation must  be  sufficiently  extensive  and  definite  to  distin- 
guish the  thing  defined  from  everything  else.  Definitions, 
in  mathematics,  are  used  to  determine  the  meaning  of  the 
terms,  as  well  as  the  signs  and  symbols  used. 

10.  An  Axiom  is  a  self-evident  truth  or  proposition.  They 
are  said  to  be  self-evident,  because,  as  soon  as  enunciated, 
they  produce  in  the  mind  a  force  of  conviction  that  cannot 
be  increased  by  any  subsequent'  train  of  reasoning.  This 
conviction  is  the  result  of  an  instantaneous  and  intuitive 
perception  of  the  simple  relations  involved. 

11.  By  a  skilful  use  of  the  simple  elements  of  mathemati- 
cal knowledge,  furnished  by  Definitions  and  Axioms,  we  are 
led  on  through  the  most  complicated  processes  of  mathe- 
matical investigation. 

DEFINITIONS. 

12.  A  problem  is  a  question  proposed  which  requires  a 
solution ;  and  the  problem  is  said  to  be  solved  when  the 
value  of  the  unknown  quantity,  involved  in  the  conditions 
of  the  question,  is  discovered. 

13.  A  theorem  is  a  general  truth,  which  is  to  be  proved  by 


SECT.  I.]  AXIOMS.  15 

a  course  of  mathematical  reasoning  called   a  demonstra- 
tion. 

14.  A  Itmma  is  a  subsidiary  truth  previously  laid  down,  in 
order  to  render  the  solution  of  a  problem,  or  the  demon- 
stration of  a  theorem,  more  easy. 

15.  A  proposition  is  a  common  name,  applied  indifferently 
to  problems,  theorems,  and  lemmas. 

16.  A  corollary  is  an  obvious  consequence,  derived  from 
some  proposition  already  demonstrated,  without  the  aid  of 
any  other  proposition. 

17.  A  scholium  is.  a  remark  made  on  one  or  several  pre- 
ceding propositions,  to  point  out  their  connexion,  their  use, 
their  restriction,  or  their  extension. 

18.  A  hypothesis  is  a  supposition  made  either  in  the  enun- 
ciation of  a  proposition  or  in  the  course  of  demonstration. 

AXIOMS. 

19.  The  following  is  a  list  of  mathematical  axioms.  The 
list  is  incomplete,  but  sufficiently  extensive  for  our  present 
purpose. 

1.  The  whole  of  a  quantity  is  greater  than  a  part.   ' 

2.  Quantities  equal  to   the  same  quantity  are    equal  to 
each  other. 

3.  If  to  equal  numbers  equals  be  added,  the  sums  will  be 
equal. 

4.  If  from  equal  numbers  equals  be  subtracted,  the   re- 
mainders will  be  equal. 

5.  If  equal  numbers  be  multiplied  by  equals,  the  products 
will  be  equal. 

6.  If  equal  numbers  be  divided  by  equals,  the  quotients 
will  be  equal. 

7.  If  the  same  quantity  be  added  to  and  subtracted  from 
another,  the  value  of  the  latter  will  not  be  altered. 

8.  If  a  quantity  be  multiplied  and  divided  by  a  number, 
its  value  will  not  be  altered. 

9.  If  equal  numbers  are  involved  to  equal  degrees,  their 
powers  will  be  equal. 


16  ELEMENTS    OF   ALGEBRA.  [SECT.  I. 

10.  If  corresponding  roots  of  equal  numbers  be  taken, 
they  will  be  equal. 

11.  If  to  unequal  numbers  equals  be  added,  the  greater 
will  give  the  greater  sum. 

12.  If  from  unequal  numbers  equals  be,  subtracted,  the 
greater  will  give  the  greater  remainder. 

13.  If  unequal  numbers  be  multiplied  by  equals,  the 
greater  will  give  the  greater  product. 

14.  If  unequal  numbers  be  divided  by  equals,  the  greater 
will  give  the  greater  quotient. 

ALGEBRAIC    NOTATION. 

20.  Jllgehra  is  that  branch  of  mathematical  science  in 
which  the  relations  of  quantities  are  investigated,  and  the 
value  of  unknown  quantities  determined,  by  means  of  let- 
ters and  signs.* 

21.  Quantities^  in  Algebra,  are  represented  by  the  letters 
of  the  alphabet  as  well  as  by  numbers. 

22.  The  first  letters,  as  a,  6,  c,  &c.,  are  used  to  represent 
the  known  quantities,  'the  last  letters,  as  a?,  y,  &c.,  are 
used  to  represent  the  unknown  quantities. 

23.  The  use  of  letters  to  represent  quantities  is  product- 
ive of  several  important  advantages. 

1.  A  letter  may  be  made  to  represent  the  unknown  quan- 
tity whose  value  is  sought,  and  then  be  used  in  the  so- 
lution of  the  problem  as  though  its  value  were  already 
determined. 

2.  The  long  and  tedious  processes  of  arithmetic  may  be 
greatly  abridged  by  the  introduction  of  letters,  since  a 
single  letter  may  be  made  to  represent  any  quantity, 
however  great  it  may  be. 

3.  The  several  quantities  which  enter  into  the  calcula- 
tion are  preserved  distinct  from  each  other,  in  all  their 
combinations. 

4.  The  requisite  operations  may  be  performed  with  much 
more  readiness,  and  with  less  liability  of  mistake,  with 
letters  than  with  numbers. 

*  See  Note  B. 


SECT.  I.]  ALGEBRAIC  NOTATION.  17 

5.  The  processes  of  algebra  may  be  used  tp  demonstrate 
theorems  and  general  rules,  inasmuch  as  a  letter  mjay 
represent  every  possible  value. 

24.  The  relations  of  quantities,  or  the  operations  to  be  per- 
formed upon  them,  are  represented  by  signs.  This  method 
of  notation  presents  to  the  eye,  at  one  view,  the  conditions 
of  the  problem,  and  at  the  same  time  facilitates  the  reduc- 
tion of  it. 

25.  Addition  is  represented  by  a  horizontal  and  perpen- 
dicular line  mutually  bisecting  each  other,  as  +.  Thus, 
a-\-b  represents  that  b  is  to  be  added  to  a,  and  the  expres- 
sion is  read  "  a  plus  &." 

26.  Subtraction  is  indicated  by  a  horizontal  line  prefixed 
to  the  quantity  to  be  subtracted,  as  — .  Thus,  a — b  repre- 
sents that  6  is  to  be  subtracted  from  a,  and  the  expression 
is  read  "  a  minus  i." 

27.  Multiplication  is  indicated  by  a  sign  formed  some- 
thing like  a  Roman  X,  as  X.  Thus,  aXb  indicates  that  a 
is  to  be  multiplied  by  b.  Sometimes  the  multiplication  is 
indicated  by  a  dot  placed  between  the  quantities  to  be  mul- 
tiplied, as  a.b  ;  or  if  the  quantities  are  represented  by  let- 
ters, the  letters  may  be  written  one  after  another,  in  alpha- 
betical order,  without  any  sign,  as  ab.  If  numerals  are  to 
be  multiplied,  the  sign  must  be  expressed.  Thus,  4x  10,  or 
4.10,  without  the  sign,  would  become  410. 

28.  Division  is  indicated  in  three  ways.  1.  By  connect- 
ing the  divisor  to  the  dividend  by  a  horizontal  line  with  a 
dot  above  and  another  below  it,  as  —.  Thus,  a-rb  indicates 
that  a  is  to  be  divided  by  b.  2.  By  making  the  dividend  the 
numerator,  and  the  divisor  the  denominator  of  a  vulgar 
fraction,  as  ^.  3.  Or  by  placing  the  divisor  to  the  right  of 
the  dividend,  and  drawing  a  perpendicular  line  between 
them,  and  a  horizontal  line  under  the  divisor,  as  a\b. 

29-.  To  indicate  that  the  difference  between  two  quantities 
is  to  be  taken  without  determining  which  is  to  be  subtracted, 
a  sign  like  the  letter  s  placed  horizontally  is  used,  as^  c/t* 

C 


18  ELEMENTS    OF    ALGEBRA.  [sECT.  I. 

Thus,  a  (/)  5  represents  that  the  difference  between  a  and  h 
is  to  be  taken. 

30.  Equality  between  two  quantities  or  sets  of  quantities 
is  indicated  by  two  horizontal  lines,  as  =r.  Thus,  a=6  rep- 
resents that  a  is  equal  to  Z*,  and  is  read ,"  a  equals  6." 

31.  An  equation  is  the  algebraic  expression  of  two  equal 
quantities  connected  by  the  sign  of  equality.  If  the  alge- 
braic quantities  are  known,  the  expression  is  called  an 
equality. 

•  32.  Inequality  is  indicated  by  two  lines  forming  an  angle, 
like  the  letter  V  placed  horizontally,  the  vertex  denoting  the 
less  of  the  two  quantities,  as  >.  Thus,  ayb  represents 
that  a  is  greater  than  &,  and  is  read  "  a  greater  than  Z)." 

33.  An  inequation  is  the  algebraic  expression  of  two  un- 
equal quantities  connected  by  the  sign  of  inequality.  If 
the  quantities  are  known,  the  expression  is  called  an  ine- 
quality. 

34i.  Proportion  is  indicated  in  the  same  manner  as  in  Com- 
mon Arithmetic.  Thus,  aib'.'.cd  represents  that  the  four 
quantities  a,  Z>,  c,  and  c/  are  proportional,  and  the  expression 
is  read  "  a  is  to  6  as  c  is  to  c?." 

35.  A  coefficient  is  a  numeral  figure  or  a  letter  prefixed 
to  a  quantity  to  show  how  many  times  the  quantity  is  to  be 
taken.  Thus,  4a  shows  that  a  is  to  be  taken  four  times,  as 
o+a+a+a=:4a;  and  ax  shows  that  x  is  to  be  taken  as 
many  times  as  there  are  units  in  a. 

36.  When  a  quantity  has  no  number  prefixed  to  it,  1  is 
always  understood  as  its  coefficient.  Thus,  a  is  the  same 
as  la. 

37.  An  Mgehraic  expression  is  a  quantity  or  several  quan- 
tities written  in  algebraic  language ;  that  is,  by  the  aid  of 
letters  and  signs. 

39.  An  algebraic  formula  is  a  general  rule  or  principle 
stated  in  algebraic  language ;  that  is,  by  the  aid  of  letters 
and  signs. 

39.  A  monomial  or  simple  algebraic  quantity  is  one  that 


SECT.  I.]  ALGEBRAIC    NOTATION.  19 

may  be  represented  in  an'Rlgebraic  expression,  without  the 
aid  of  the  signs  pltts  or  minus.  Thus,  a,  3a6,  iab',  and  lab^mx 
are  monomials.     Monomials  are  sometimes  called  terms. 

40.  Polynomials^  or  compound  quantities,  are  expressions 
containing  two  or  more  simple  quantities  <;onnected  by  the 
signs  plus  or  minus.  Thus,  a+3a6  and  a-\-*Xb — 3c  are  poly- 
nomials. 

41.  A  polynomial  composed  of  two  terms  is  called  a  W- 
nomial ;  of  three  terms,  a  trinomial  y  of  four  terms,  a  quadri- 
nomial.  If  the  two  terms  of  a  binomial  are  connected  by 
the  sign  minus,  it  is  sometimes  called  a  residual. 

42.  To  indicate  that  like  operations  are  to  be  performed 
upon  all  the  terms  of  a  polynomial,  they  must  be  included 
in  a  parenthesis,  or  have  a  bar  or  vinculum  drawn  over 
them.  Thus,  a—(b-{-c)  indicates  that  the  sum  of  b  and  c  is 
to  be  subtracted  from  a;  and  (a-|-6)Xc  indicates  that  the 
sum  of  a  and  b  is  to  be  multiplied  by  c  ;  and  (^a-^b)^c  indi- 
cates that  the  sum  of  a  and  b  is  to  be  divided  by  c. 

43.  If  both  multiplicand  and  multiplier,  or  dividend  and 
divisor,  are  polynomials,  each  should  be  included  in  a  pa- 
renthesis, as  (a4-6)x(c-}-c/),  or  (a-|-6)H-(c+c/).  And  in 
general,  when  a  sign  is  prefixed  to  a  parenthesis,  it  is  to  be 
understood  as  affecting  all  the  terms  included  in  the  paren- 
thesis, taken  collectively. 

44.  Positive  or  additive  quantities  are  those  to  which  the 
sign  plus  is  prefixed.  J^egative  or  subtractive  quantities 
are  those  to  which  the  minus  sign  is  prefixed.  When  no 
sign  is  prefixed  to  the  first  term  of  an  algebraic  expression, 
the  sign  plus  is  always  to  be  understood. 

45.  A  quantity  is  said  to  be  ambiguous  with  regard  to  its 
sign  when  it  is  affected  with  the  double  sign  ± .  Thus, 
a±b  represents  that  b  is  to  be  added  to  or  subtracted  from 
a;  and  the  expression  is  read  "a  plus  or  minus  A." 

46.  Equal  terms  affected  by  unlike  signs,  in  an  algebraic 
expression,  cancel  each  other,  and  may  be  rejected  from  the 
expression.  Thus,  3a — 56+5^=3«,  since  — 5^  and  -f56 
cancel  each  other. 


20  ELEMENTS    OF    ALGEBRA.  [sECT.  I. 

47.  Positive  and  negative  quantities  sustain  opposite  rela- 
tions with  respect  to  addition ;  i.  e.,  a  negative  quantity 
must  be  subtracted  when  a  positive  quantity  would  be  addp 
ed,  and  added  when  a  positive  quantity  would  be  subtracted. 

48.  The  numbers  which  are  multiplied  together  to  form 
a  composite  number,  are  called /ac^'or^.  Thus,  Wabcx  is  a 
composite  number,  formed  by  multiplying  the  factors  11, 
a,  ^,  c,  and  X. 

49.  A  number  is  said  to  be  resolved  into  factors  when  two 
or  more  numbers  are  taken,  such  that,  when  multiplied  to- 
gether, their  product  shall  equal  the  given  number.  Thus, 
54  may  be  resolved  into  6x9,  or  3x18,  or  2x27. 

50.  The  power  of  a  number  is  the  product  arising  from 
the  multiplication  of  the  number  by  itself,  till  it  has  been 
used  as  a  factor  a  certain  number  of  times.  If  the  number 
is  taken  twice  as  a  factor,  the  product  is  called  the  second 
power  ;  if  three  times,  the  product  is  called  the  third  power  ; 
if  four  times,  the  fourth  power,  &c.. 

51.  The  index,  or  exponent,  is  a  figure  or  letter  placed  to 
the  right  and  a  little  above  the  number,  and  is  used  to  show 
the  power  to  which  the  number  is  to  be  involved.  The 
number  is  to  be  used  as  a  factor  as  many  times  as  there  are 
units  in  the  exponent.  When  no  exponent  is  expressed,  1 
is  understood. 

The  first  power  of  a  is         »        -         -        «,  or  a\ 
The  second  power  of  a  is  -         -    axa,  or  a^. 

The  third  power  of  a  is      -         -       axaxa,  or  a-^. 
The  fourth  power  of  a  is     -  axaxaxa,  or  a'^. 

The  mth  power  of  a  is         ax  ax  a  m  times,  or  a",  &c-.. 

52.  If  a  polynomial  is  to  be  involved,  its  terms  should  Be 
included  in  a  parenthesis,  and  the  exponent  placed  without 
the  parenthesis  to  the  right.  Thus,  (a-]-b)-  i«  the  algebraic 
expression  of  the  second  power  of  the  sum  of  a  and  b* 

53.  Involution  is  finding  the  powers  of  numbers. 

54.  The  root  of  a  number  is  a  number  which,  multiplied 
into  itself  till  it  is  taken  a  certain  number  of  times  as  a 


SECT.  I.]         ALGEBRAIC  NOTATION.  21 

factor,  will  produce  the  given  number.  The  root  is  called 
square  root,  cube  root,  fourth  root,  &c.,  according  to  the 
number  of  times  it  must  be  used  as  a  factor  to  produce  the 
given  number. 

55.  The  radical  sign,  as  ^/  ,  or  fractional  index,  is  used 
to  indicate  that  the  root  of  a  number  is  to  be  taken.  The 
denominator  of  the  fractional  index  denotes  the  root ;  and 
when  the  radical  is  used,  the  figure  over  the  foot  of  the 
radical  determines  the  root.     Thus, 

The  square  root  of  a  is  expressed         -         \/a,  or  a*. 
The  cube  root  of  a  is  expressed  -         y/'a^  or  ai. 

The  fourth  root  of  a  is  expressed         -         ^ a,  or  oi 
The  fifth  root  of  a-\-b  is  expressed,  \/a-\-b,  or  {a-\-b)\,  &c. 

56.  Evolution  is  finding  the  roots  of  algebraic  numbers. 

57.  A  power  of  a  root,  or  root  of  a  power,  is  a  result  ob- 
tained by  involving  the  root  of  a  number,  or  by  extracting 
the  root  of  a  power.  Cases  of  this  kind  are  indicated  as 
follows : 

The  second  power  of  the  third  root  of  a  is  v'a*,  or  (^a)*i 

or  a!. 
The  third  power  of  the  fourth  root  of  o-f-6  is  ^(a-|- J)*,  or 

(a+b)i. 

58.  It  should  be  remarked  that  the  exponent  aflfects  only 
the  letter  over  which  it  is  placed.  Thus,  in  the  expression 
abc^,  the  first  powers  of  a  and  by  and  the  second  power  of  c, 
are  to  be  taken.  When  no  coefficient  is  prefixed  to  tho 
radical  sign,  1  is  always  understood  as  the  coefficient. 

59.  Exponents  should  not  be  confounded  with  coefficients. 
The  exponent  indicates  that  the  number  is  to  be  used  as  a 
factor  a  certain  number  of  times.  Thus,  o"  represents  that 
a  is  to  be  taken  six  times  as  a  factor,  or  axaxaxaxaxa 
=0*.  The  coefficient  indicates  that  the  number  is  to  be 
used  as  a  term  a  certain  number  of  times.  Thus,  6a  repre- 
sents that  a  is  to  be  used  six  times  as  a  term,  or  a-|-a-|-a-h 
a4-fl+a=6a. 

60.  The  reciprocal  of  a  quantity  is  the  quotient  arising 


22  ELEMENTS    OF   ALGEBRA.  [sECT.  I. 

from  dividing  a  unit  by  that  quantity.     Thus,  the  reciprocal 

of  a  is  7j  of  a-\-h  is  ^ip^  ;  and  of  4  is  J . 

61.  The  reciprocal  of  a  power  is  the  quotient  arising  from 
dividing  a  unit  by  that  power,  and  is  frequently  expressed 
by  a  negative  exponent.     Thus, 

The  reciprocal  of  a^  is        -         -         -       ^,  or  a~^. 
The  reciprocal  of  4a^  is      -         -  4;^,  or  \a~^. 

The  reciprocal  of  {a-\-hy  is         -  (^a-\-b)^t  or  (a+i)~^,  &c. 

62.  Rational  quantities  are  those  whose  exact  value  can  be 
expressed  in  finite  terms.  Thus,  4a,  ^6,  and  o-|-3Z>,  are  ra- 
tional quantities. 

63.  Irrational  quantities^  or  surds,  are  those  whose  exact 
value  cannot  be  expressed  in  finite  terms.  Thus,  since  only 
the  approximate  value  of  the  square  root  of  2  can  be  ob- 
tained, s/2  is  called  a  surd  j  also  Va  is  a  surd. 

64.  The  measure  or  divisor  of  a  quantity  is  thai  by  which 
it  can  be  divided  without  leaving  a  remainder  ;  and  when  a 
quantity  will  divide  two  or  more  quantities  without  leaving 
a  remainder,  it  is  called  a  common  measure  of  those  quanti- 
ties. Thus,  la  is  a  measure  of  28a,  since  7^=4 ;  and  3a  is 
a  common  measure  of  12a  and  21a,  since  —  =4,  and  ^1^=7. 

65.  The  multiple  of  a  quantity  is  that  which  can  be  divided 
by  the  quantity  without  leaving  a  remainder.  Thus,  28a  is 
a  multiple  of  7a,  since  ^=4,  &c. 

66.  Commensurable  quantities  are  such  as  have  a  common 
measure  or  divisor.  Thus,  12a  and  21a  are  commensurable, 
because  they  have  a  common  divisor,  3  or  3a. 

67.  Incommensurable  quantities  are  such  as  have  no  com- 
mon measure  except  unity.  Thus,  5  and  7,  3a  and  10^,  are 
incommensurable  quantities. 

68.  The  value  of  an  algebraic  expression  is  the  result  ob- 
tained by  substituting  for  the  letters  their  numerical  values, 
and  performing  the  operations  indicated  by  the  signs.  Thus, 
the  value  of  4a— 8Z>,  on  the  supposition  that  a=i:12  and  ^=:5, 
is  4  X  12—8  X  5=48— 40  =  8. 

The  value  of  la+s,  on  the  supposition  that  a=:6  and^= 
10,  isix6+^l°=3+6=9. 


SECT.  I.]  ALGEBRAIC  NOTATION.  23 

69.  The  following  examples  are  given  for  the  exercise  of 
the  learner.  On  the  supposition  that  a=6,  6=5,  c=4fj  d=. 
1,  and  m=10,  it  is  required  to  find  the  value  of  the  follow- 
ing algebraic  expressions. 

1.  a^-\-<2ab+b'=6^+2  .  6  .  5+5^=36  +  60+25=121. 

2.  2a'— 3a'6+c»=2.6'— 3.6^5+4»= 

3.  (a+Z») xa^— 5c(//7i+V'  =  (6  +  5).  6^— 5  .4  . 1 .  10+^= 

4.  4^+a.(2c+c»)— 3w  = 

5.  (4.c'+6').a— (6+3A^).8= 

7.  5v/c%^~a*.(3a'— IOot— 7</)= 

8.  ^T.(3a+6+4m)— 6a.(36+(f)= 

9.  3.  v/4^c:3^+3a.(2a  +  6— (i)i= 
10.  b  .  Va*-\-'6d^m — 3bc\/b^^= 

1  1       2H-g         ^&fc=^Wf-t-ct 

*A'     3a-«  2o+c     —      

12.  (-;;;zr-+at>)i— — ^ —  = 

70.  The  value  of  an  algebraic  expression  is  not  altered 
by  changing  the  order  of  the  terms  or  the  order  of  the  fac- 
tors, if  its  proper  sign  be  prefixed  to  each.  Thus,  a+6+c — 
d  is  the  same  in  value  as  — rf+c+ft+a;  and  aXbXcxd  is 
the  same  in  value  as  dxcxbxa.  For  considering  the  let- 
ters of  the  same  value  as  in  art.  69,  we  shall  have  a+6+ 
c— <i=6  +  5+4r-l=— l+4  +  5+6  =  14j  and  aXbXcXd= 
6.5.4.1  =  1.4.5.6=120. 

Mote. — It  will  be  found  convenient  to  write  the  letters  in 
alphabetical  order. 

71.  Like  quantities  or  terms  are  those  which  consist  of 
the  same  letters  and  the  same  powers,  or  the  same  roots  of 
the  same  letters.  Thus,  3a,  6a,  5a  are  like  quantities ;  and 
7a*6,  8a'6,  and  a  b  are  also  like  quantities. 

72.  Unlike  quantities  or  terms  are  those  which  consist  of 
different  letters,  or  dififerent  powers  of  the  same  letters. 
Thus,  3a  and  Sab  are  unlike  quantities,  because  they  have 
different  letters ;  and  3a  and  3a^  are  unlike,  because  differ- 
ent powers  of  the  same  letter  are  taken. 

73.  A  polynomial  which  is  composed  of  like  quantities  or 


24  ELEMENTS    OF    ALGEBRA.  [sECT.  I. 

terms  may  be  reduced  to  one  term.  Thus,  ba-\-3a—Sa ; 
for,  letting  a— 6,  and  substituting  for  a  its  value,  the  expres- 
sion will  become  5.6 -{-3  .  6=  8.  6,  or  30+18=:48.  Again, 
5a — 3a=2a;  for,  letting  a=z6,  and  substituting,  as  before, 

5  .  6—3  .  6  =  2  .  6,  or  30— 18=  12.  In  the  third  place,  —5a 
— 3a= — 8a;  for  substituting,  as  before,  — 5.6 — 3.6=: 
— 8.6,  or  — 30 — 18= — 48.  The  same  method  of  illustra- 
tion may  be  applied,  whatever  value  we  assume  for  a. 

JVote. — The  addition  of  negative  quantities  may  seem  in- 
consistent at  first  sight ;  but  it  should  be  recollected  that  the 
minus  sign  merely  indicates  that  the  quantities  affected  by 
it  are  to  be  subtracted ;  hence  the  sum  or  aggregate  of 
those  quantities  must  also  be  subtracted.  Thus,  if  a  mer- 
qhant  loses  $30  in  one  speculation  and  $18  in  another,  the 
sum  of  his  losses  is  $48,  and  the  algebraic  expression  of  it, 
is  —$30— 18=— $48. 

74.  A  polynon^ial  which  is  composed  of  unlike  quantities 
or  terms  cannot  be  reduced  to  a  simpler  form.  Thus,  Sa-f- 
3o  can  be  reduced  to  no  simpler  expression  ;  for,  letting  a=: 

6  and  6=5,  we  shall  have  5a+36=5  .  6  +  3  .  5=45  ;  but  5a+ 
3b  cannot  equal  8a,  since  8  .  6=48 ;  neither  can  it  equal  86, 
since  8  .  5=40  :  therefore  the  polynomial  cannot  be  reduced 
to  a  simpler  expression. 

75.  The  processes  of  algebra  are  employed  chiefly  in  the 
solution  of  problems,  or  in  the  demonstration  of  theorems 
and  the  investigation  of  general  rules.  This  is  accomplished 
by  means  of  a  series  of  equations  or  proportions.  But,  be- 
fore entering  upon  the  consideration  of  these,  it  will  be 
necessary  to  make  an  application  of  the  algebraic  method 
of  notation  to  the  fundamental  principles  of  numeration. 


SECT.  II.]  ADDITION.  25 


SECTION  II. 

Additioriy  Subtraction^  Mrdtiplication^  and  Division  of 
Algebraic  Quantities* 

ADDITION. 

76.  Addition  is  a  method  of  finding  the  sum  of  two  or 
more  algebraic  quantities. 

77.  This  may  be  done  by  connecting  the  several  quanti- 
ties by  their  proper  signs  in  one  algebraic  expression. 

78.  In  order  to  bbtain  the  simplest  expression  for  the  sum 
of  two  or  more  quantities,  it  is  necessary  to  reduce  the  like 
quantities  to  one  term.  Accordingly,  Addition  may  be  con- 
veniently considered  in  three  cases. 

CASE   I. 

79.  In  this  case  the  quantities  are  like,  and  have  like 
signs. 

RULE. 

1.  Write  the  quantities  to  be  added  so  that  the  like  terms  may 
fall  under  each  other, 

2.  jJdd  the  coefficients^  and  to  their  sum  prefix  the  common 
sigUj  and  annex  the  common  letter  or  letters. 

JSTote. — For  the  reason  of  this  and  the  next  rule,  see  the 
illustrations  in  Art.  73. 

EXAMPLES. 


(1) 

3a+  2A—  5c 

5«4-  eb—    c 

^a+llb—Sc 

a+     b—  3c 

(2) 

4a6 —  2cd 

lab—     cd 

l^ab—  2cd 

ab—12cd 

(3) 

la%x+\2c}^ 
%a%x-\-  Scl/ 
2(^bx^     c6» 
Sa^bx+  IcV" 

I6a+20fr— 17c 
3 

Tiab-llcd 
D 

20a«^x-f.28ci» 

26  ELEMENTS    OF   ALGEBRA.  [sECT.  II. 

4.  Add  Sa'—^bc,  4a'— 56c,  la'—Sbc,  and  ^a'—Sbc. 
Arranging  the  terms  for  addition,  3a" — 26c 

4a' — 56c 
7a'— 86c 
2a'— 36c 


5.  Add  —Sax—2by\  —2ax—6by%  —3ax—by\  —Qax-^^Sby^ 
and — 3aj? — 6/.  ^ns. — 25aa7 — 186y'. 

6.  Add  12a6'c— 8c</a7'+56/,  a6'c— 7crfa:^-|-36y^  and  3a6'c-- 
cc?ar'+26/.  Jlns.  16a6'c— 16cJa?^+ 106y\ 

7.  Add  3a6  +  2aa7y,  7a6+4aa?y,  and  12a6-|-10aa:y. 

^ns.  22a6-j-16axy. 

8.  Add  Ho^^'y— 3a?y+4a?/,  ex'y—l2xY+6xf,  and  ar'y— 
3xY-}-xy\  Ans.  24a?='y—18a;'y' +110:3/'. 

9.  Add  8a'6='c^— 36'a?^  12a'6V— 66'a:^  13a'63c^— 76V,  2a'6V 
— 66'a:^  and  lla'6V— 136'a?^  Ans.  46a'6V— 356'a:\ 

10.  Add  60a6— 12(a+6),  30a6— 3(a+6),  40a6— 7(a+6), 
and  80a6— 3(a-f-6).  Ans.  210a6— 25(a+6). 

CASE    II. 

80.  In  this  case  the  quantities  are  like,  but  the  signs  un- 
like. 

RULE. 

1.  WriU  the  quantities  to  be  added  so  that  the  like  terms  may 
fall  under  each  other. 

2.  Take  the  difference  between  the  sum  of  the  coefficients  of 
the  positive,  and  the  sum  of  the  coefficients  of  the  negative 
terms,  and  to  this  difference  prefix  the  sign  of  the  greater  sum, 
and  annex  the  common  letter  or  letters, 

EXAMPLES. 

(1)    '  (2)  (3) 

12a6+6aa?  7a'a7-l-13a6y  6cc^l2a:y 

— 7a6+3aa?  Qa^x-\-     ab^y^  cd —  3xy 

3ab—1ax  — 3a'a?—  7a6y  — 3cc/-|-   ^xy 

8a6+2aa7  10a'5;+  7a6'/  4c(/—  Qxy 

4.  Add  2a— Sor^,  7a+5a?'  — 3a+a?',  and  a  +  3a?'. 

Ans,  7a +60:' 


SECT.  II.]  ADDITION.  27 

5.  Add  4a6V— 12tir,  —3ab'c'-\-Sdx,  lab'c'+6dx,  — 3a6V— 
Idx,  Ans,  5aZ»-c*— 5cir 

6.  Add  7aAz*— ISaV+Qic",    — SoAz"— aV— 66c^    3a6z*-f 
4a*x'— 6k^  and  — a^^z^+^aV— 76c». 

^»*.  oiz*— 6a«x'-T-106c^ 

7.  Add  12a»y^4-13,  So'y*— 7,  — o^y^+S,  2ay— 9,  and  —3 
ay— 11.  Ans,  13ay— 6. 

8.  Add  6a?4-5ay,  — 3x  +  2ay,  a? — 6ay,  and  2a?4-ay. 

^;i5.  6a:4-2ay. 

9.  Add  — 3aft4-7a^6,  3a*— 10a*6,  3a*— 6a'*,  —ab—^a'b,  and 
2ab-^la-b.  Ans,  4fab — 4a**. 

10.  Add  3a(a+*),  7a(o  +  *),  — 5a(a  +  *),  and  3a(a+*.) 

Ans.  8a(a -{-*). 

11.  Add  7(6a?-fy— 2)',  _8(6x+y— 2;)^  (Gx+y- z^,  and 
3(6x+y— 2)^  ^715.  3(6x+y— 2)^ 

12.  Add  3a*+4a(6y4-*),— 8a*— 9a(6y-f-*),  12a*+13a(6y 
+  *),  ab+a(6y-^b),  and  7a*+6a(6y+*). 

^iw.  15a*+15a(6y+*). 

CASE    III. 

81.  In  this  case  the  quantities  are  unlike,  or  some  like 
and  others  unlike. 

RULE. 

1.  Reduce  the  like  terms  as  in  the  preceding  cases.- 

2.  To  the  results  thus  obtained^  connect  the  terms  which  con- 
not  be  reduced  by  their  proper  signs, 

J^ote. — This  rule  is  founded  on  the  principles  illustrated 
in  articles  73  and  74. 

EXAMPLES. 

1.  Add  3ay^  — 2a:y',  — 3y'x,  — 8a?'y,  and  2Ty^. 

f  3ay'— 2xy- 
These  terms  may  be  arranged  thus :  <         — 3xy' — 8a?*y 

I         +2xy' 

3ay  — 3j:y  — Sor'y 


,*. 


V 


28  ELEMENTS    OF    ALGEBRA.  [sECT.  11. 

2.  Add  8oV,  — Sax,  lax,  — 6xy,  — bax,  9xy,  2a V,  and  xy, 

rSaV— 3aa? 


These  terms  may  be  arranged  thus  : 


-\-lax — 6xy 

— ^ax-\-9xy 

2o^a^  +  xy 


lOaV  —  ax-\-^xy 
3.  Add  10b^—Sa\  — 5H2aV,  ^0-{-2a%  and  aV+120 

^ns.  9b^-{-3a'x'—a'x+110, 
4>.  Add  ab-^Sy  cd~3,  28,  and  5crf— 4m  +  2. 

^ns.  ab-\-6cd-\-3b — 4>m, 

5.  Add  babXc'cP,  labXc"^,  6abcd,  and  —3abXc'd^-j-3abcd. 

Ans.  9ab  x  c'd^^9abcd. 

6.  Add  3aa7— 21,  65c-f2,  aa;+15— 56c,  — 8  +  6ca:— 6c,  and 
llaa;+ 13— 36c.  ^ws.  21aa;— 36c+ 1. 

7.  Add  3m2— 1,  Gam— 4m^+8,    7— 9aOT+8,    Gm^— 3-f  aw, 
and  4>m^ — am  +  12.  Ans.  ^rr^ — Sam +31. 

8.  Add   \Sx{a^—b^\   —60^0?+ 1262a:,   —  10a: (0^—60 +13aX 
and  Sx{a^—V)  +  36^0;.  ^?i5.  6a;(a2— 6-)  +  7a'a:+  156^3;. 

9.  Add  4a2+36+2c,  — 3a2  +  46  +  8c,  Qa^— 76— 10c,  and  3a» 
— 6  +  c+aa7.  Ans.  \Sa — 6+c  +  oa;. 

>10.  Add  8aa?+2(a?+a)+36,  9aa;+6(a:+o)— 96,  and  — 7aa;— 
8(a;+a)+66+lla:.  ^»*.  10aa:+lla?. 

11.  Add  5a26+12(a— a:)^  3a''6— 8+9(a— a:)^  12— 8a26,  and 
--13(a— a:y+3.      .  Ans.  8(a— a;)-+7. 

12.  Add  28aXa7+5i/)+21,  —  13a''(a:+5y)+18«,  — 15a'(a:+ 
5y)_8,  and  —13— 8a.  Ans.  10a. 

>  13.  Add72aa:^— Say',— 38aa:^—3ay^+7a3^,  8+ 12a3^^— 6a/ 
4, 12— 34aa:'+  baf—^ay\  Ans.—2ay^-^  20. 

14.  Add    12a— 13a6+16aa:,   8— 4ffi+2y,  — 6a+7a62+12y 
—24,  and  7a6— 16aa:+4m.    Ans.  6fl— 6a6+  14y+7a6'— 16. 

15.  Add   17a(a:+3a2/)+12a='6V,    8— 18ay— 8a^6V— 7a(a:+ 
3a2/),  — 4+12a?/— 10a(a:+3a2/)— 4a'6V\and  6ay— 4.  Ans.  0. 

16.  Add  Sab's^—Sa'cd,  -^7a6V+7a='cc/— 12,   32a6V,    and 
12-Sa^cd+ab'x\  •  Jlns.  29a6V— 40=^0^ 


gECT.  II.]  SUBTRACTION.        *^,  29 

SUBTRACTION. 

82.  Subtraction  is  a  method  of  finding  the  difference  be- 
tween two  algebraic  quantities  or  sets  of  quantities.  It  is 
the  opposite  of  Addition. 

83.  We  have  already  seen  that  a  negative  quantity  is  of 
an  opposite  nature  to  a  positive  quantity  (Art.  4>7),  with 
respect  to  addition  and  subtraction :  that  is,  it  must  be  sub- 
tracted when  a  positive  quantity  would  be  added,  and  added 
when  a  positive  quantity  would  be  subtracted.  Hence,  for 
subtraction  of  algebraic  quantities,  we  have  the  following 
general 

RULE. 

1.  Write  the  quantity  to  he  subtracted  under  that  from  which 
it  is  to  be  taken,  placing  like  terms  under  each  other, 

2.  Change  the  signs  of  all  the  quantities  to  be  subtracted,  or 
conceive  them  to  be  changed^  and  then  proceed  as  in  addition* 

EXAMPLES. 

(1)  (2)  (3) 

From  Qa—db       Uab—  6ac'— 12aj:y      Scd?—8axy-i-3bd 
Take    3a — 4.6         bab~10ac'-—  2axy     Ucd^—  axy+3bd 

Ans.  3a— 56         6a6+  4.ac^— tOaxy  — 3cc/^ — laxy 
4.  From  3a6^— 8aVa:'+26"c  take  2a6^+4a^cr'4-3A*c. 

^/w.  a6^-12a'cr'— 6V. 

♦  The  principles  on  which  this  rule  is  founded  may  be  stated  and  de- 
monstrated as  follows  I 

1.  Subtracting  a  positive  quantity  will  produce  the  same  result;  as  add- 
ing an  equal  negative  quantity. 

Represent  the  sum  of  two  quantities  by  -       -       fl-j-* 

Taking  -f^  away  from  this  expression,  there  remains  a 

Adding  — b  to  it,  we  shall  have  ...        a-j.^ — fc=at 

2.  Subtracting  a  negative  quantity  will  produce  the  same  result  as  add- 
ing an  equal  positive  quantity. 

Represent  the  difference  of  two  quantities  by        -        a — h 

Taking  — b  away  from  this  expression,  there  remains  a 

Adding  -|-A  to  it,  we  shall  have  -       -       .       a — i-|J-*c=i. 


30  ELEMENTS  OP  ALGEBRA.  [sECT.  II. 

5.  From  12ax—19c^b+2abx—a^  take  dax—Qc'b—Sabx. 

^ns.  '^ax—10c%+\0abx—3^ 

6.  From  ^a^b—llcd'^+^y—^aT?  take  —^a'b—10cd^-\-^y— 
bax^^2cd\  Ans.  la'b—Sax". 

7.  From  ISa'd-^xy-^-d  take  la^d—xy-^-d-i-hm^—ry. 

jSns.  Qa!^d-{-2xy — hm'^-\-7^y'^, 

8.  From  '7a^bc^—S-\-'7x  take  3a^bc'—S—da^-^r. 

Ans.  ^a^he+nx+dx'—r. 

9.  From  l^a'bx^^llb-^c'  take  \\a'bo^^9b—lc\ 

Ans.  la'bx^+^b—c'. 

10.  From  IGa^iV— 34-48c/a^  take  — 4a25V+&a:+12c(/. 

Ans.  '^0a^Wx^—3—bx-\-^Ux—l%cd, 

11.  From  3a5c— 8a;2/+25H-85take  —  lla&c+4a;y— 22— 7J. 

^;i5.  14ak— 12i:y+4.7+15J. 

12.  From  the  sum  of  6a?^y — llajj^and  8ar^y+3aa;^  take  4a?^y 
— ^aar*.  ./^W5.  lOir^y— 4aar'. 

13.  From  the  sum  of  15ak4-8ccic — 3  and  24— 8«Z?c+2ccte 
take  the  sum  of  12aJc — 3cdx — 8  and  — 4aJc+cc?a;+16. 

Ans.  — dbc^Vilcdx-\-\3, 
14.^  From  the  difference  between  8a& — 12ca;  and  — 3ab-{- 
4iCx  take  the  sum  of  bab — lex  and  ab-\-cx. 

Arts.  5ab — lOcx. 
15.  From  the  sum  of  4aa;''+2ar'+350,  5aa;'+6a;'+250,  and 
9aa^+ 12a?=*+ 100,  take  the  sum  of  6aa;2+9a:'+432,  a3^-{-5x^-^ 
328,  and  5aa^+a:3+30.  Ans.  6ax''+bx''—90. 

84.  The  minus  sign,  when  placed  before  the  marks  of  pa- 
renthesis which  inchide  a  polynomial,  indicates  that  each 
term  of  the  polynomial  is  to  be  subtracted,  or  that  the  result 
obtained  by  reducing  the  terms,  if  they  are  like  quantities, 
is  to  be  subtracted.  This  is  done  by  removing  the  marks 
of  parenthesis,  and  changing  the  signs  of  the  terms  included 
between  them.     Thus: 

1.  3a— (3c— a?) = 3a— 3c + a?. 

2.  Sabc — {laic + 3a?— 5)  =  3a Jc— 2a Jc— So? + & = a^c— 3a7  -f-  J. 

3.  4aa? — 3c— 14— (aa?+ 7c  — 12):=: 4aa: — 3c— 14— oa?— 7c+ 
12r=3aa?— lOc— 2. 


SECT.  II.]  MULTIPLICATIOrf,  31 

4.  7abc  —  13-\-Sabx—(3alc--U-\-9ahx)=labc-^13-^Sabx-^ 
3abc-\-  14j— 9aix=4.aic-f  1 — abx. 

85.  When  a  number  of  terms  are  introduced  within  the 
marks  of  parenthesis,  to  which  the  minus  sign  is  prefixed, 
the  signs  of  the  terms  should  be  changed.     Thus : 

1.  3ax—12a^3b=3ax—(-\-12a+3b). 

2.  3abc—Q-\'4'ab+3x=3abc—{6—^ab^3x), 

3.  lxy—12ab—4y—S—b=lxy—{nab+^y-{-S-\-b). 

86.  By  the  above  methods,  polynomials  may  be  made  to 
undergo  a  variety  of  transformations,  which  are  sometimes 
of  great  use  in  algebraic  operations. 

87.  The  word  ^ddiiion^  as  here  used,  it  will  be  perceived 
from  the  foregoing  operations,  does  not  always  imply  in- 
crease or  augmentation,  nor  does  the  word  Subtraction  al- 
ways imply  diminution.  Hence  the  term  Reduction  has  been 
sometimes  employed  to  express  the  operations  included  un- 
der addition  and  subtraction. 

MULTIPLICATION. 

88.  Multiplication  is  repeating  the  multiplicand  as  many 
times  as  there  are  units  in  the  multiplier.     Thus : 

1.  If  a  is  to  be  multiplied  by  by  it  must  be  taken  as  many 
times  as  there  are  units  in  b,  and  the  expression  would 
become  o  x  i,  or  ab, 

2.  If  ab  is  to  be  multiplied  by  erf,  it  must  be  taken  as  many 
times  as  there  are  units  in  cd^  and  the  expression  would 
become  abxcd,  or  abed.  Hence,  to  multiply  letters^  we 
write  them  one  after  the  other ^  in  alphabetical  order. 

d>  If  4a  is  to  be  multiplied  by  36,  it  must  be  taken  as  many 
times  as  there  are  units  in  3h.  Thus,  4ax3Z*=4'X<2x3 
X  6=4  X  3  X  a  X  i  —  \^ab.  Hence,  numerical  coefficients 
must  be  multiplied  together ^  and  their  product  prefixed  to  the 
product  of  the  letters. 

4.  If  la"  is  to  be  multiplied  by  4a',  it  must  be  taken  as 
many  times  as  there  are  units  in  4a',  and  the  work  may 
be  opressed  thus:    7a'x4a'=7aax4aaa=7xaax4x 


32  ELEMENTS    OF   ALGEBRA.  [sECT.  II. 

aaa—lx^Xaaxaaa  —  ^'^a^—^^a^+^.  Hence,  if  the  same 
letter  is  found  in  both  factors,  it  is  multiplied  by  adding  to- 
gether its  exponents,  and  their  sum  is  the  index  of  the  same 
letter  in  the  product. 

89.  With  regard  to  the  signs,  it  should  be  observed,  that 
if  the  signs  of  the  two  factors  are  like,  the  sign  of  the  product 
will  be  +  j  but  if  their  signs  are  unlike,  the  sign  of  the  pro- 
duct will  be  — .     This  rule  may  be  illustrated  thus: 

1.  If  +«  is  to  be  multiplied  by  -{-b,  the  multiplication  con- 
sists in  repeating  -\-a  as  many  times  as  there  are  units 
in  -i-b  J  and,  consequently,  the  product  is  -\-ab. 

2.  If  — a  is  to  be  multiplied  by  -{-b,  the  multiplication 
consists  in  repeating  — a  as  many  times  as  there  are 
units  in  -\-b  ;  and,  consequently,  the  product  is  — ab. 

3.  If  -^a  is  to  be  multiplied  by  — b,  the  minus  multiplier 
indicates  that  the  repetitions  of  -\-a  are  to  be  sub- 
tracted i  consequently,  the  product  is  — ab. 

4.  If  — a  is  to  be  multiplied  by  — b,  the  repetitions  of  — a 
will  be  negative  ;  but  the  minus  multiplier  indicates 
that  these  repetitions  are  to  be  subtracted  j  conse- 
quently, the  product  is  -\-ab. 

90.  It  should  also  be  remarked,  that  if  there  are  more  than 
two  factors,  an  odd  number  of  negative  factors  will  produce 
— ,  and  an  even  number  -\-.  Thus,  — ax — bx — c= — abc  j 
for  — ax — b=-\-ab,  and  -{-abx — c= — abc.  Again,  — aX 
— bx — ex — d=-\-abcd ;  for  — ax — h=-]-ab,  and  -\-abx — c 
= — abc,  and  — abcx — d—-\-abcd. 

91.  The  classification  of  quantities  into  monomials  and 
polynomials  suggests  three  cases  in  multiplication,  viz. : 
When  the  factors  are  both  monomials ;  when  one  is  a  poly- 
nomial and  the  other  a  monomial  j  and  when  both  are  poly- 
nomials. 

CASE    I. 

92.  In  this  case  the  factors  are  monomials,  and  the  signs 
are  like  or  unlike. 


SECT  II.]  MULTIPLICATION.  33 

RULE. 

1.  Multiply  together  the  numejncal  coefficients  of  the  facUrrs. 

2.  To  the  product  of  the  coefficients  annex  the  product  of  the 
letters^  observing  that  if  a  letter  is  contained  in  both  the  multl- 
plican(t  and  multiplier^  it  will  be  affected  mith  an  exponent  in 
the  product  equal  to  the  sum  of  its  exponents  in  the  factors. 

3.  Prefix  to  the  product  the  sign  required  by  the  principle 
that  like  signs  produce  pluSy  and  unlike  signs  minus. 

JVote. — For  an  illustration  of  the  principles  Df  this  rule, 
see  Arts.  88  and  89. 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.) 

Multiply  lOa^bx      —  laxyz  12t/m'»V         —l^a'bc'd'i^ 

By  3a&V  3abcx        —  Sd^fmn  —  Sabx 

Product  lo^l^'  ^2Wbcx'yz  —96d]fm'n'x*     +39o'^Vrf*x' 

(5)  (6)  (7) 

Multiply       Sa*h'<^d^a^y*      —  HoVyz*         —  ^Sa^z 
By  la'bc'd'xY  12  —  l^a'b'cdJ'xyz^ 

Product      36a'6Vc/Vy'      — 204aVyz*        +336a'b'cd'xy'2^ 

8.  Multiply  lla^bcdhy  4<a^b*cx,  jJns.  Ua'b'c^dx, 

9.  Multiply  iSamxy  by  6anxz.  Jins.  lOSa^mnx'yz. 

10.  Multiply  ^eahdJ'y  by  —2ac'df.     Ans.  — 192aVciy. 

11.  Multiply  —31abcd  by  babcx.         ^ns.  — IGOa^^^Vt/x. 

12.  Multiply  — 12a6  by  —U^ab\  Ans.  +1728a*^»*. 

13.  Multiply  6o*i'  by  Vlc^xy,  and  that  product  by  2oar'/. 

Ans.  144a«&»a?y. 

14.  Multiply  Wbc  by  3aZ>V,  and  that  product  by  —1c?b(?. 

Ans.  — 48a«i>V. 

15.  Multiply  — la^x^f  by  ^ac^xf,  and  the  product  by  — 4a- 
xf.  Ans.  +64aVy'. 

16.  Multiply  — lOJc*  by  — 3^>*cV,  and   that  product  by 
— 4tVxy.  Ans.  — 120Z;'cVy. 

17.  Multiply  the  product  of  ISr'y  and  5a:^y-  by  the  product 
of  xY  and  3xy.  Ans.  270a?"y\ 

18.  Multiply  IZa'Vcd'  by  10a«ft»a:. 

E 


34  ELEMENTS    OF    ALGEBRA.  [SECT.  II. 

19.  Multiply  ^la'b'c'd'x  by  —12adnxK 

20.  Multiply  — 16aV  by  —16. 

21.  Multiply  6ada  by  3a,  and  that  product  by  12a^JV. 

22.  Multiply  — lax  by  ISaix,  and  that  product  by  — 12 
a'ccy, 

23.  Multiply  —6c^x'z'  by  — 12a='a?,  and   that  product  by 
—2ax'z\ 

24.  Multiply  the  product  of  Kax  and  3axz  by  the  product 
of  8ak^and  2aV/. 

25.  Multiply  the  product  of  ISaz  and  2a^a?  by  the  product 
of  6aa?  and  — da'^y^z'^, 

26.  Multiply  the  product  of — Sax  and  — 12aa?  by  the  pro- 
duct of  — 4<ax  and  Sax. 

27.  Multiply  Sahcd  by  12aV,  and  that  product  by  —Sa^b^ 
c^dx. 

28.  Multiply  the  product  of  — Sahx  and  4<a^xz  by  the  pro- 
duct of  6z  and  lla^lfx^z'^. 

29.  Multiply  lla^d'  by  — SaV*,  and  that  product  by  lWd\ 

30.  Multiply  the  product  of  —ISahH'  and  — 12aV(^V  by 
the  product  of  —Sa'c'd'xY  and  — 7a'cMVyV. 

CASE    II. 

93.  In  this  case  the  multiplicand  is  a  polynomial  and  the 
multiplier  a  monomial. 

RULE. 

1.  Multiply  the  letters  and  coefficients  of  each  term  of  the 
multiplicand  by  the  letters  and  coefficients  of  the  multiplier, 

2.  Prefix  to  each  term  of  the  product  the  sign  required  by  the 
principle  that  like  signs  produce  plus^  and  unlike  signs  minus. 

J^ote. — The  simple  principle  on  which  this  rule  rests  is,  that 
the  sum  of  all  the  units  in  the  multiplicand  is  to  be  taken  as 
many  times  as  there  are  units  in  the  multiplier.  Thus,  if 
a-\-b  is  to  be  multiplied  by  c,  it  is  evident  that  both  a  and  h 
must  be  taken  as  many  times  as  there  are  units  in  c  ;  hence, 
{a-\-b)xc—ac-\-bc. 


SECT.  II.] 


MULTIPLICATION. 


35 


EXAMPLES. 

(1) 

Multiply       ^ah+cd 
By  Sac 

Product       12a'bc-\-3ac'd 

(3) 
Multiply      a-{-3b—2c 

By  —3bx 

Product  — 3abx  —  9b'^x-{-6bcx 

(5) 
Multiply    3a^xY—l-\-a 
By  5(^xy 

Product 


(2) 

6abd 


l^a'bd—lSab'd 

(4) 
2b— la— 3 
4<ab 
8ai^—2Sa^b—12ab 

(6) 
—12a''bx'—4fbc' 
—  3a 

36a'bx'-\-12abc^ 


15a  Vy' — oa^xy  -f  6a*xy 

7.  Multiply  lla*Z>c'— ISxy  by  3ax, 

Ans.  33a%c^x—3^a:^y, 
8._  Multiply  42c^— 1  by  —4.  .y?;w.  — 168c^+4. 

9.  Multiply  — 30a'fta?V+ 13  by  —So*. 

^715.  +150a55a;2y— 65a^ 

10.  Multiply  the  product  of  a-\-b  and  3c  by  8ax. 

.^?w.  24a^ca:+24a&ca7. 

11.  Multiply  the   product    of   2a-\-3b — 4c   and  — 2a  by 
%abdx.  Am.  —32o?bdx—^>'^a^b^dx-\-^^>a%cdx, 

12.  Multiply  the  sum  of  3aJH- 10  and  a2>— 8  by  6aa?. 

Ans.  24a'*a?4-12aa?. 

13.  Multiply  the  sum  of  12aJx — %ad — 3b  and  %abx — ad-\-b 
by  3a.  Arts.  60a''bx—27a'd—6ab. 

14.  Multiply  the  difference  between  12a — Ibdx  and  8a+ 
bdx  by  6abdx.  Ans.  2Wbdx—4,Sab^<Px', 

15.  Muhiply  the  sum  of  16— 3y+12k  and  y +10— bd  by 
2abdm.        Ans.  52abam — ^abdmy-\-24fab^cdm — 2ab^^m. 

16.  Multiply  20a'^»V/— l+17aJy4-3x2  by  —^a'h^<^d\ 

Ans,  — 180a'iVc/Vy + 9a<J2c^J^—  153a'*6V(^*y— 

^la'b^c'd'xz. 

CASE   lU. 

94.  In  this  case  the  multiplicand  and  multiplier  are  both 
polynomials. 


36  ELEMENTS  OF  ALGEBRA.  [sECT.  II. 

RULE. 

1  Multiply  the  letters  and  coefficients  of  each  term  of  the 
multiplicand  by  the  letters  and  coefficients  of  each  term  of  the 
multiplier, 

2.  Prefix  to  each  term  of  the  product  the  sign  required  by  the 
principle  that  like  signs  produce  plus,  and  unlike  .sig?is  minus. 

Jiote  1. — The  principle  on  which  this  rule  is  founded  is, 
that  in  multiplication  the  sum  of  the  units  in  the  multipli- 
cand is  to  be  taken  as  many  times  as  is  expressed  by  the 
sum  of  the  units  in  the  compound  multiplier.  Thus,  if  a-{-h 
is  to  be  multiplied  by  c-\-d,  it  is  evident  that  a-\-b  is  to  be 
repeated  as  many  times  as  there  are  units  in  c-{-d ;  hence, 
since  c-\-d  cannot  be  reduced  te  a  monomial,  we  repeat  the 
multiplicand  c  times,  and  then  d  times,  and  then  take  the 
sum  of  the  repetitions.  Thus,  a~\-b  repeated  c  times  gives 
ac-}-bc,  and  a-\-b  repeated  d  times  gives  ad-\-bd ;  and  the 
sum  of  the  repetitions  is  ac-\-bc-\-ad-^bd. 

Jfote  2. — Like  terms  in  the  product  should  be  placed  un- 
der each  other,  and  the  product  reduced  to  its  simplest  form. 


(1) 

ultiply  2a  +3& 
Y               a-\-   b 

EXAMPLES. 

6a?y 
3ax 

(2) 

—2^               < 
—^d 

2a' -f-  3ab  \^ax^y—Qaxz—3Qdxy+  IQdz 

2aZ»+3&2 


Product    2a2+5a&+362 

(3) 
Multiply           a-\-b  +c 
By                     a —  b — c 

a^^ah-\-ac 
^ah        —5'—  be 
— ac       —  he — (? 

Product 

a^               ^j^^^lbc—c' 

SECT.  II.]  MULTIPLICATION.  37 

(4) 
Multiply  Sab  . —    Vy—<d 
By  6a*  —  ^h^y-\-cd 


ISa^l^'—eab^y—eabcd 

—6aPy  -\-2b*y-\r2cb^dy 

+  3abcd  —  cVdy—c^d} 

Product  \Sa}y^l1ab^y—Zabcd+^b*y^  cb^dy—c^<j^ 

5.  Multiply  &+C+2  by  J+c+3. 

Jlns.  62+2Jc+5J4-c*+5cH-6. 

6.  Multiply  2a+3b-\-c  by  ia+c+l. 

^ns.  8a"4-12aft+6ac-f3Jc+c*+2a4-3&+c. 

7.  Multiply  a^+ft^  by  a+b,  Jhis.  cr'+ai'+a^ft+J*. 

8.  Multiply  m+^bc+ld"  by  3i«+2c*. 

./fns.  9J«+15J'c+27^c2-f  10J<r»+14c\ 

9.  Multiply  a-f-^>  by  a—b,  ^ns.  a^—V, 

10.  Multiply  2a— i  by  3a'— 1.      .^w*.  6a»— 3a'fr— 2a+J. 

11.  Multiply  3ai*— 6  by  a+4. 

Ans.  Sa^J'- 6a-f-12a&«— 24. 

12.  Multiply  6a4-46  by  3a— 2Z».  Ans,  18a«— 86«. 

13.  Multiply  7aZ>c^+3xy+l  by  %a%K 

Ans.  b^a^'c^  -}-  24a'^&'a:y  +  8a»**. 

14.  Multiply  4a'ftr'+  Zed  by  3ctir,  and  that  product  by  4a 
-f*.        Ans.  48a'k(ir*-f  36ac'(/2a?+  na^bhdx'-\-Uc^d'x. 

15.  Multiply  a-\-b-\-c  by  8aZ>,  and  that  product  by  a-\-b. 

Ana.  8ff'64-  16a«6'+8a-6c  +  8a6'-f-8a6»c. 

16.  Multiply  2a4-46  by  2a— 46.  Ans.  ^a^—\U\ 

17.  Multiply  ar'+x^y-f-xy^+y'  by  x — y.  Ans.  x* — y*. 

18.  Multiply  x^-faryH-y^  by  x^ — <sy-\-y^'  Ans,  x*-^f-{-y*. 

19.  Multiply  a:y+l  by  3a-f-*,  and  that  product  by  4c. 

Ans.  12aci'y-|-12ac-f-46ca;y-f-46c. 

20.  Multiply  3b-\-2x-^h  by  3ax6x2cx2x. 

Ans.  S6ab''cx + 24aica:'  + 1 2abchx. 

21.  Multiply  3j:»+2a?Y-t-3y'  by  2x'— Sr^y^+S/. 

Ans.  ex"— 5xy'+ 6x»y3— 6x^*4-  ISx^y'+xy-j-  15y". 

22.  Multiply  a'^^b^-^-x'—ab—ax—bx  by  a-f  6+a:. 

Ans.  a'— 3a*a?+6'4-a?». 


38  ELEMENTS    OF   ALGEBRA.  [sECT.  11. 

23.  Multiply  a?*+a:y +y^  by  x'—y'.  Ans.  x^—y\ 

24.  Multiply  together  a-{-b,  a^-^ab-\-h^,  a—b,  and  a'—ab^ 
b\  Ans.  a'—b'. 

95.  For  many  purposes,  it  is  sufficient  to  indicate  the  mul- 
tiplication of  polynomials,  as  {a-{-b)x(a-\-b-\-c) ',  and  when 
the  multiplication  is  performed,  the  expression  is  said  to  be 
expanded. 

96.  As  the  multiplier  merely  expresses  the  number  of 
times  the  multiplicand  is  to  be  repeated,  it  is  always  con- 
sidered a  number. 

97.  The  multiplicand  may  be  either  a  quantity  or  a  num- 
ber ;  and,  since  repeating  a  quantity  cannot  change  its  na- 
ture, the  product  will  be  of  the  same  nature  as  the  multipli- 
cand. 

98.  We  sometimes  speak  of  multiplying  dollars  by  yards 
or  pounds ;  but  this  language,  if  construed  literally,  is  absurd. 
To  obtain  the  cost  of  a  given  number  of  articles,  we  repeat 
the  cost  of  one  article  as  many  times  as  there  are  articles 
purchased. 

99.  If  the  multiplier  is  a  unit,  the  product  will  be  equal  to 
the  multiplicand ;  if  it  is  greater  than  a  unit,  the  product 
will  be  greater  than  the  multiplicand ;  but  if  it  is  less  than 
a  unit,  the  product  will  be  less  than  the  multiplicand.  And, 
in  general,  with  this  same  multiplicand,  the  product  de- 
creases as  the  multiplier  decreases  ;  and  if  the  multiplier  be 
reduced  to  0,  the  product  is  0.  Hence,  if  0  enters  as  a  fac- 
tor into  any  quantity  whatever,  the  value  of  the  expression 
becomes  0. 

DIVISION. 

100:  Division  is  finding  a  quotient  which,  being  multiplied 
into  the  divisor,  will  produce  the  dividend.  It  is  the  con- 
verse of  multiplication,  the  product  and  one  of  the  factors 
being  given  to  find  the  other  factor. 

101.  It  is  evident,  from  the  nature  of  division,  that  a  fac- 
tor equal  to  the  divisor  must  be  rejected  from  the  dividend  ; 


SECT.  II.]  DIVISION.  39 

or,  the  coefficient  of  the  dividend  must  be  divided  by  the 
coefficient  of  the  divisor,  and  a  factor  equal  to  the  literal 
part  of  the  divisor  rejected  from  the  literal  part  of  the  divi- 
dend.    Thus,8aZ»^4ft=:^=2a;  for  Ux2a^Sab. 

102.  If  the  same  letter  is  found  in  both  dividend  and  divi- 
sor, and  affected  with  a  greater  exponent  in  the  dividend 
than  in  the  divisor,  the  exponent  of  the  quotient  will  be 
that  of  the  dividend  diminished  by  that  of  the  divisor. 
Ihus,  a/  —  a" ^zaaaaaaa— aaaaa=z—^i^i;^ z=aa=^ar zzzo''^, 

103.  If  the  exponent  of  the  divisor  is  greater  than  the  ex- 
ponent of  the  same  letter  in  the  dividend,  the  exponent  of 
the  quotient  will  be  negative.     Thus,  a^H-a^=a*~''=:a'l. 

If  the  divisor  contains  a  letter  that  is  not  found  in  the 
dividend,  the  exponent  of  that  letter  will  be  aff^ected  with 
the  opposite  sign  in  the  quotient.  Thus,  a'"-T-J'"=a'"i~"' j 
for  a"'6-"*x6"'=a"'6-'"+'"=:a'"6°=a'".  (Art.  107.) 

104.  The  same  rule  is  observed  with  regard  to  the  signs 
in  division  as  in  multiplication,  i.  e.,  like  signs  produce  plus, 
and  unlike  signs  minus.     Thus, 

H-a6-f-4-6=  a;  for  +ax+6=+a6. 
-\-ab-r- — b= — a;  for — aX — b=-\-ab. 
— ab—-\-b= — a ;  for  — ax  -\-b= — ab. 
— ab-r- — b=-\-a;  for  -fax — b— — ab, 

105.  The  operations  to  be  performed  in  division  may  also 
be  conveniently  considered  in  three  cases,  accordingly  as 
the  quantities  are  both  monomials,  or  as  the  dividend  is  a 
polynomial  and  the  divisor  a  monomial,  or  as  both  are  poly- 
nomials. 

CASE   I. 

106.  In  this  case,  the  dividend  and  divisor  are  both  mo- 
nomials. 

BULE. 

1.  Divide  the  coefficient  of  the  dividend  by  the  coefficient  of  the 
divisor. 

2.  Reject  the  letters  common  to  both  dividend  and  divisor  when 
they  have  the  same  exponent;  but  when  the  exponents  are  not 


40  ELEMENTS    OF    ALGEBRA.  [sECT.  II. 

the  same^  subtract  the  exponent  of  the  divisor  from  that  of  the 
dividend^  and  the  remainder  will  be  the  exponent  of  that  letter  in 
the  quotient. 

3.  To  the  above  results,  annex  the  letters  of  the  dividend  that 
are  not  found  in  the  divisor,  and  also  those  of  the  divisor  that 
are  not  found  in  the  dividend,  observing  that  the  signs  of  the  ex- 
ponents of  the  latter  are  to  be  changed. 

4.  Prefix  to  the  whole  the  sign  required  on  the  principle  thai 
like  signs  produce  plus,  and  unlike  signs  minus. 

EXAMPLES. 

(1)                    (2)  (3)               (4) 

Divide      Sa-bd        —2Sa'c'dxY  ^baW            ^Wb&d 

■    By            2a6                  Ic^cHxY  5«'^       —36 

Quotient  ^ad           —  ^xy  la~^b~^        —  ^a^bc^d 

5.  Divide  na'b^x  by  Zab\  Ans.  Ubx. 

6.  Divide  48a''cV^a;V  by  Sa'd^ar'.  Ans.  ^ac'dzK 

7.  Divide  42cfc/a?^  by  — ladx.  Jins.  — 6a?^. 

8.  Divide  12aWc''x  by  ^a^b'x.  Jlns.  ^a'bc". 

9.  Divide  120a:y  by  — Sa'^y.  Jlns.  —Ibxy, 

10.  Divide  —846c'  by  —126c.  Jlns.  +7c. 

11.  Divide  256a'6V  by  8a^6V.  Ans.  32a'b'c-\ 

12.  Divide      56a^6W  by  SaWdxK  Ans.  la^bH^x, 

13.  Divide  90a'6cMa?y    by    5a'bc'd'x'y. 

Ans.  ISa-'c-'d-'x-y, 

14.  Divide  98a^6^  by  —4>9a'bK  Ans.  —2abK 

15.  Divide  620j?y  by  4^xYz\  Ans.  lb5xy'z-\ 

16.  Divide  —15a'b'  by  ba'b'c'd.  Ans.  —3aWc-'d-\ 

17.  Divide  m^a'b-'d'  by  12a'6Vc?'.       Ans.  U4>a-^b'c-^d\ 

18.  Divide  684a?'  by  — 12a?-^  Ans.  —57a?'. 

19.  Divide  328007^^^  by  4>0xyz\  Ans.  82a7yV. 

20.  Divide  62a'6'  by  31a-'b-^d\  Ans.  2a'b'd-\ 

21.  Divide  the  product  of  Sa'bd'  and  laWdx,  by  2Sa'bdx, 

Ans.  2aWd^. 

22.  Divide  the  product  of  ^a'b'c'd'x'  and  —lOabc'd  by  —2a' 
^►V.  An^.  -h25ac^(^V. 


SECT.  II.]  DIVISION.  41 

23.  Divide  the  sum  of  12a*AV  and  8a^6V  by  the  sum  of 
3a6V+7aZ>V.  J^ns.  2a^bc, 

24.  Divide  the  difference  of  21a'iV(/-V  and  15a^b*c^(Px*  by 
6a'bcdx,  JJns.  aU'c^da^. 

107.  It  sometimes  occurs  in  the  operations  of  division, 
that  a  letter  becomes  affected  with  the  exponent  0.  We 
will,  therefore,  explain  that  symbol.  Thus,  a^-ra'=a*~*=:a° ; 
or,  again,  a"'-ra"*=a'"^^=a°  j  but  ^=1,  and  °m=lj  there- 
fore, since  a  may  represent  any  quantity  whatever,  and  m 
any  exponent  whatever,  every  quantity  affected  with  the  ex- 
ponent 0  is  equal  to  1. 

This  will  also  explain  why  a'^^h'"—arh-'^  (Art.  103)  j  for, 
multiplying  the  divisor  and  quotient  together,  &"*xa'"i~'"= 

CASE    II. 

108.  In  this  case  the  dividend  is  a  nolynomial  and  the 
divisor  a  monomial. 

KULE. 

1.  Divide  each  term  of  the  dividend  by  the  divisor^  and  the  re- 
sulting quantitieSy  connected  by  their  proper  signs,  will  be  the 
quotient. 

JVote. — That  each  term  of  the  dividend  should  be  divided 
by  the  divisor,  is  evident  from  the  fact  that  when  a  polyno- 
mial is  multiplied  by  a  monomial  factor,  that  factor  enters 
into  every  term  of  the  polynomial.  Thus,  (a-^b-\-c)xd= 
ad+bd+cd;  hence,  (ac/+W+c(f)^c/-'5^±^:^^+?+?=:a-f 
i+c. 

EXAMPLES. 

(1)  (2)  (3) 

Divide      2ab+6bc        12a?'y4-39aV/  42aVx+ ISflWo?' 

By  2b  3x'y  —Qac'x 

Quotient    a  -{-3c  4  4.13ay  —la'    —  3 Wo?* 

(4)                            (5)  (6) 

Divide      12a'b'c—21a''b'c*         lDaxy-{.12cd         9aWx—3abc- 
By  3abc' 3 3ab(^ 

Quotient    ^-^bc-'—lb'c'  5axy+^cd  3aJVx— 1 

F 


42  ELEMENTS    OF   ALGEBRA.  [sECT.  II. 

7.  Divide  lUd'b^c'^Sc^b'c''  by  8a'/>V.  ^ns.  9abc—(^, 

8.  Divide  35dm-\-Udx  by  Id.  Jlns.  ^m-\-1x, 

9.  Divide  4aa:y— 4a+16ac?  by  4«.  Ans,  a?y— l4-4c?. 

10.  Divide  3aa:='+6:c'+3aa;-15a?  by  3x. 

Ans.  007^4- 2a: + a— 5. 

11.  Divide  3a'bc-\-l2abx—3a%  by  3aJ.      c/^7^.s.  c+A>x—a, 

12.  Divide  25a^j£c— 15axa;^4-5aJc  by  —  5aa:. 

Ans.  — bab-\-3acx — bcx~\ 

13.  Divide  20ah^+15ab'+10ab-^5a  hy  5a. 

^?i5.  4&^+3'^+2S+l. 

14.  Divide  the  product  of  9a%^-\-la'x'  and  4aVJ-  by  2a^b. 

Ans.  lSa^c''d'-{-Ud'b~'c'd'x'. 

15.  Divide  the  product  of  12a'^^a;='+3Ja:'  and  8a:'  by  24<bx\ 

Ans.  4>a^x^-\-x^. 

16.  Divide  the  product  of  Gaxy-^-lc^bc^dx  and  6x'^-\-4>'if  by 
2aa?.  Jgns.  lSx'y+21a^c'dx^-{-12y'-{Ua^bc'df, 

CASE    III. 

109.  In  this  case  the  dividend  and  divisor  are  both  poly- 
nomials. 

RULE. 

1.  Arrange  the  terms  of  the  dividend,  and  also  those  of  the 
divisor,  with  reference  to  the  power  of  some  letter,  so  that  its  ea?- 
ponents  shall  diminish  from  left  to  right. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor  ;  the  result  is  the  first  term  of  the  quotient. 

3.  Multiply  the  whole  divisor  ly  this  term,  and  subtract  the 
product  from  the  dividend  ;  the  remainder  will  form  a  new  divi- 
dend. 

4.  Divide  the  first  term  of  this  new  dividend  by  the  first  term 
of  the  divisor  ;  the  result  is  the  -  second  term  of  the  quotient. 
Multiply  the  divisor  by  this  term,  and  subtract  as  before. 

5.  Proceed  in  this  manner  till  the  dividend  has  been  exhausted, 
or  till  no  term  of  the  remainder  is  divisible  by  the  first  term  of 
the  divisor. 

JVote  1. — In  applying  the  above  rule,  we  find,  successively, 
how  often  the  divisor  is  contained  in  parts  of  the  dividend, 
for  the  reason  that,  as  the  dividend  is  made  up  of  all  its  parts. 


SECT.  II. J  DIVISION.  43 

the  divisor  is  contained  in  the  whole  as  often  as  it  is  con- 
tained in  all  its  parts. 

JVb/e  2. — If  the  first  term  of  the  dividend  is  not  divisible 
by  the  first  term  of  the  divisor,  after  the  terms  in  each  have 
been  arranged,  the  division  is  impossible. 

J^ote  3. — If  the  dividend  is  exactly  divisible  by  the  divi- 
sor, the  dividend  will  be  completely  exhausted,  leaving  no 
remainder.  But  if  it  is  not  exactly  divisible,  the  division 
may  be  continued  till  the  first  term  of  the  remainder  is  not 
divisible  by  the  first  term  of  the  divisor  j  the  remainder 
should  then  be  placed  over  the  divisor  so  as  to  form  a  frac- 
tion. 

J^ote  4. — It  will  not  in  all  cases  be  necessary  to  bring 
down  all  the  terms  of  the  dividend  to  form  the  first  re- 
mainder. 

EXAMPLES. 

1.  Divide  a'4-2a6+*'  by  a-\-b. 

Dividend,   a'+2oZ>+6'|a-f-6^  Divisor. 
a' 4-  ab        a+^,    Quotient. 

0    0 
Proof.  (a+6)x(a+6)=a'+2a6+6^ 

2.  Divide  12a'*6*— 6a^*'H-8o'6'— 4a»6*— 22a«i+5a^by  4a«i»-h 
5fl*— 2a'A. 

Dividend  arranged.  Divisor  arranged. 

5a^— 22a«6+12a^ft»—  6a^y— ■4a»y+8a'6^|5a^-2(r»6+4fl'y 
Sa"'—  2a%-\-  4a'6'  Quotient,  a'—4>a'b-{-2b^ 


•— 20a«6-h  8aV/—  6a^i»— 4aV+8a«6* 

— 20o^6-h  Sab'~'i6a'b' 

•  ♦  -{-lOa^b^-U'b^+Sa'b' 

lOa^b^—W'b^+Sa^b' 


0 
Proof.  (5(1*— 20^6+ 4a'i>^)  xCo*— 4a'JH-4a'J»)=5a'— 22a*'6-f 

12a^  J^— 6a*i»— 4a'i*4-  Sa'b\ 


44  ELEMENTS    OF   ALGEBRA.  [SECT.  U, 

3.  Divide  0=^— 1  by  a— 1. 

Dividend.  Divisor, 
a^—l        |g— 1 
o' — a^        a^+a+1,  Quotient. 
-fa^— 1 
a'^ — a 


+  a— 1 
a— 1 
0 
4.  Divide  a® — b^  by  a— &. 


■i-a'b^—a^b' 


~\-a'b'—b' 
a'h^—d'b^ 


-{-a'b'—ab' 


+ab'—b^ 

'    ab'—b^ 

0 

5.  Divide  a^-^-^a'b-^-^ab^-^-b^  by  a-{-b.    Ans.  ct'^^^ab^y", 

6.  Divide  a='+2a'6+2aZ''+Z'='  by  c^^ab^bK      Ans.  a+b, 

7.  Divide  x'—^x'-\-'2nx—rt  by  a?— 3„    Ans.  a?2_6a'+9. 

8.  Divide  a?^+y^  by  x^y.  Ans.  x'^—xy+y\ 

9.  Divide  6a=^a?~9aV— a^+4a?*  by  a^+Sa?^— 3aa?. 

^7i5.  2x'^-\-3ax—a^. 

10.  Divide  6aa7+2a?2/ — Sab — by-\-3ac-{-cy  by  Sa+y. 

11.  Divide  2a?^--19a?2+26a?— 16  by  a?— 8. 

Ans.  2a?'— 3a?+2. 

12.  Divide  3/'+ 1  by  y4- 1.  ^^^5.  y^—f+y^—y+ 1- 

13.  Divide  /—I  by  y—1.      Ans.  y'+y'+y'+^+y+l- 

14.  Divide  a?^ — a'  by  x—a.  Ans.  x+a. 


SECT.  II.]  DIVISION.  45 

15.  Divide  x* — a'  by  x — a.  Jlns:  ar*-|-aa?'-fa'x-|-a'. 

16.  Divide  --15a^  +  37a'*c— 29a'tir-— 206V+44.k(ic— 8J^x* 
by  Sa'—bbc+dx.  ^ns.  — 5a'H-46c— 8fir. 

17.  Divide  3a*— 8a^6'4-3aV+56'— 36V  by  a^— ^. 

Jlns.  3a2— 5d*4-3c». 

18.  Divide  20a'— 41a*6-|-50a='fr^— 4.5a-6^H-25a6*— 6/»'  by  4a' 
— 5aZ>4-2A-.  ^ns.  ba'^A^a^b+bab'—Sb^ 

19.  Divide  9a:«— 46x'-f  95x^+1 50x  by  x^—4>x—5. 

Arts,  9a:*— 10x^+5x2— 30a?. 

20.  Divide  6x*— 96  by  3a:— 6.     Ans,  2ar'+4x-+8x+16. 

21.  Divide  4324-1152Z»VH-576iV  by  6+126V. 

Ans.  72+48^>V. 

22.  Divide  8aV— 8a''6a?^4-8a''cx^— 11  a^^ +116-^—11^2  by 
o— 64-c.  ./?»5.  8aV— II62. 

23.  Divide  6x''—5xy—6xy+6ar'y'+15yx='—9xy +10x^3/* 
H-  15y'  by  3x»+  2xy + 3/.  Am.  2x''— 3x '3/-+  5^. 

24.  Divide  a''+8a'6+28a*'6«+56a'6''+70a*6*+56a^6'+28a»6« 
+806'+*'  by  a*-|-4a'6+6a-6^4-4a6»4-6\ 

^»5.  a*+4a'6+6o^6'+4a6»+6*. 

It  is  sometimes  desirable  to  resolve  a  polynomial  into  its 
original  factors.  The  principles  of  division  enable  us  to  do 
this ;  for,  having  obtained  one  of  the  factors  by  inspection 
or  trial,  the  other  may  be  obtained  by  division.     Thus, 

1.  4a6c-|-4axy-f  4a6(/=i4a(6c+xy+W). 

2.  72a'6+4a6c=4fl6(18a+c). 

3.  8a-cx— 18acx'+2ac''y— 30aVx=2ac(4ax  — 9x»+c*y— 15 
flVx). 

4.  x'+2axH-a'=(x+a)x(x+a). 

5.  X*— 2ax4-a^=(a:— fl)x(x— a). 

6.  42x''y— 28x^y  =  7xy6x-4). 

In  the  multiplication  of  compound  quantities,  when  the 
signs  are  unlike,  some  of  the  terms  disappear  or  are  cancel- 
led in  the  product.  These  terms  will  reappear  in  the  divis- 
ion, so  that  the  quotient  frequently  contains  more  terms 
than  the  dividend.     Thus, 

(a«— «ur+x")  X  (a+6)=a»+a?». 


46  ELEMENTS    OF   ALGEBRA.  [sECT.  II. 

But  {a'^-\-cc^)-^a+x=za'—ax-]-x\ 
110.  The  division  of  quantities -may  sometimes  be  carried 
ad  infinitum.     In  such  cases  it  will  be  sulSicient  to  write  out 
a  few  of  the  leading  terms.     Thus,  ,,  ' 

1         \l—x 
1 — X  l-{-x-{-x''-\-x^j  &c. 

»  -^oc 

-{-X — x^ 

^      +x' 
-^x^ — x^ 


111.  In  multiplication,  the  multiplier  is  always  considered 
a  number,  but  the  multiplicand  and  product  may  be  either 
numbers  or  quantities.  In  division,  we  have  the  product 
and  either  one  of  the  factors  to  find  the  other.  Hence,  the 
dividend  and  divisor  may  be  either  numbers  or  quantities. 

112.  If  the  dividend  and  divisor  are  both  numbers,  the 
quotient  will  be  a  number.     Thus,  12-r-4=:3. 

113.  If  the  dividend  is  a  quantity  and  the  divisor  a  num- 
ber, the  quotient  will  be  a  quantity  of  the  same  kind  as  the 
dividend.     Thus,  12  rods-r4  =  3  rods. 

114.  If  the  dividend  and  divisor  are  both  quantities,  the 
quotient  will  be  a  number.     Thus,  12  rods-^4  rods  — 4. 

115.  From  the  nature  of  division,  it  is  evident  that  the 
value  of  the  quotient  depends  upon  both  the  divisor  and 
dividend. 

If  the  dividend  be  multiplied  while  the  divisor  remains 
the  same,  the  quotient  will  be  multiplied.  Thus,  ab-T-b=za; 
but  multiplying  the  dividend  by  m,  abm-i-h:^am. 

Dividing  the  dividend,  while  the  divisor  remains  the  same, 
divides  the  quotient.  Thus,  aim^h=am  ;  but,  dividing  the 
dividend  by  m,  ah^h=a. 

If  the  divisor  be  multiplied  while  the  dividend  remains 


SECT.  Ill  ]  ALGEBRAIC    FRACTIONS.  47 

the  same,  the  quotient  will  be  divided.  Thus,  abm-rb=am  ; 
but  abm-7-b7n=a. 

Dividing  the  divisor,  while  the  dividend  remains  the 
same,  multiplies  the  quotient.  Thus,  abm-r-hmziza ;  but  abm 
—b=am. 

116.  The  student  will  observe  that  there  is  a  striking  re- 
semblance between  the  division  of  compound  numbers  in 
algebra,  and  what  is  termed  "long  division"  in  common 
arithmetic.  But  this  essential  difference  should  be  noted  ; 
the  several  terms  are  so  independent  of  each  other,  that  af- 
ter the  first  term  of  the  quotient  has  been  obtained,  and  the 
first  remainder  brought  down  for  a  new  dividend,  an  en- 
tirely new  arrangement  of  the  terms,  with  reference  to  a 
different  letter  from  that  first  assurfied,  may  be  made  in  both 
the  divisor  and  dividend,  and  the  division  completed  under 
this  new  arrangement  without  afl^ecting  the  value  of  the 
quotient. 


SECTION  III. 

•Algebraic  Fractions. 

REDUCTION  OF  ALGEBRAIC  FRACTIONS. 

117.  Algebraic  Feactions  are  perfectly  analogous  to  vul- 
gar fractions  in  common  arithmetic.  They  express  a  part 
or  parts  of  a  whole  number,  or  unity. 

118.  The  denominator  shows  the  number  of  parts  into 
which  the  unit  is  divided  j  the  numerator  shows  how  many 
of  these  parts  are  taken. 

119.  Every  case  in  division  may  be  expressed  in  a  frac- 
tional form,  the  dividend  being  used  as  the  numerator,  and 
the  divisor  as  the  denominator. 

120.  The  denominator  and  numerator,  taken  together,  are 
called  terms  of  the  fraction. 


48  ELEMENTS    OF    ALGEBRA.  [sECT.  lU. 

121.  A  pr op kJ"  fraction  is  one  whose  numerator  is  less  than 

its  denominator.     Example, ^Zl. 

a-\-b 

122.  An  improper  fraction  is  one  whose  numerator  is  equal 

to,  or  greater  than,  its  denominator.     Example,  SjI-. 

a — b 

123.  A  mixed  number  is  an  integer  or  whole  number  con- 
nected with  a  fraction  by  the  sign  plus  or  minus.     Exam- 

ple,.a+-r 
c 

124.  A  compound  fraction  is  the  fraction  of  a  fraction,  the 
simple  fractions  of  which  it  is  composed  being  connected 

by  the  word  of     Example,  _  of  -. 

b       d 

125.  The  value  of  a  fraction, is  the  quotient  resulting  from 
the  division  of  the  numerator  by  the  denominator.  Hence, 
if  the  numerator  equal  the  denominator,  the  value  of  the 
fraction  is  a  unit ;  if  the  numerator  is  less  than  the  denomi- 
nator, the  value  is  less  than  a  unit  j  and  if  the  numerator  is 
greater  than  the  denominator,  the  value  is  greater  than  a 
unit. 

126.  The  principles  involved  in  the  reduction  of  Algebraic 
Fractions  are  the  same  as  those  applied  in  arithmetic.  It 
will,  however,  be  necessary  to  trace  out  those  operations  in 
accordance  with  the  method  of  notation  adopted  in  Algebra. 

CASE    I. 

DISCUSSION  OF  SIGNS. 

127.  The  sign  that  is  prefixed  to  the  horizontal  line  drawn 
between  the  numerator  and  denominator,  determines  whether 
the  value  of  the  fraction  is  to  be  added  or  subtracted. 

128.  A  sign  prefixed  to  one  of  the  terms  of  the  numerator 
or  denominator  affects  only  that  term. 

129.  If  the  ^\gTi  prefixed  to  the  fraction  be  changed  from  + 
to  —  or  from  —  to  +,  the  value  of  the  fraction  will  also 
be  changed  from  +  to  — ,  or  the  contrary. 

Thus,  + — —-{-ab^b  —  a  ;  but — — = — ab~b=  —  a, 
b  b 


SECT.  III. J         ALGEBRAIC  FRACTIONS.  49 

Again,  4-^t^=(a^-hac)-«=4-ft+c;  but-f*±^=-. 
a  a 

((ai-|-ac)-ra)=— (A+c)=— 6— c. 

And,  ^±Z^=(ab-ac)--a=b^c;  hut  --±I^=^(iab 
a  a 

— ac)^a)~ — (6— c)=— A-hc. 

130.  If  the  sign  prefixed  to  the  several  terms  of  the  nuvterator 
be  changed  from  -f-  to  —  or  from  —  to  -|-,  the  value  of  the 
fraction  will  be  changed  accordingly. 

Thus,  ±^=-f-a5-f-a=:6;  but  11^=— a6-^a=-^. 
a  a 

Again,  e^±ff=(aA-f  ac)--a=A+c,-  but  =±=^=(-afr- 
OL  a 

ac)-r-a=—b—c. 

And,  — —Z^  =--((a5— ac)H-a)=— (J— c)= — h-{-c;  but 
a 


a 

131.  If  the  sign  prefixed  to  the  several  terms  of  the  denomina" 
tor  be  changed  from  +  to  —  or  from  —  to  +»  the  value 
of  the  fraction  will  be  changed  accordingly. 

Thus,  —=ab-i-a=bi   but  — =ab-r- — a= — b, 
a  —a 


Again,  ±tJ^=b+ci  hat'tt^^-b-c. 
a  — a 


a  — fl 

4-c)=+fr— c. 

132.  If  any  two  of  the  above  changes  are  made,  the  value 
of  the  fraction  will  not  be  altered. 

1.  If  the  signs  before  the  fraction  and  also  before  the 
several  terms  of  the  numerator  be  changed  from  -f-  to 
—  or  from  —  to  +»  the  value  of  the  fraction  will  re- 
main the  same. 

Thus,  ^=&;  and  -3^=— (—06 -r a) =—(—6)  =+6. 
a  a 

5  6 


60  ELEMENTS    OF   ALGEBRA.  [sECT.  III. 

Again,   ?*±^=J+c;,and-=:?*=ff=:-(-J_c)=+J 
a  a 

+c. 

Also,^^-5-c;  and-=±t^=_(-6+c)r=+^^. 
a  a 

2.  If  the  signs  before  the  fraction  and  before  the  several 
terms  of  the  denominator  be  changed  from  +  to  —  or 
from  —  to  +,  the  value  of  the  fraction  will  remain  the 
same.  ^ . 

Thus,  ^=h;  and  ——  =—(ab-^—a)=—{—b)  =  -j-b. 
a  — a 

Again,  ^^=h+ci  and  -?^?-'=-((aJ+«c)-r-a)= 
a  — a 

—(^—b—c)  =  +b+c. 

Also,       ~^  =b — c;  and — ^      ^  — — ((ab — ac)-i- — a)  = 
a  — ct 

^(—b-\-c)=:-\-b—C. 

3.  If  the  signs  before  the  several  terms  of  both  nuiperator 
and  denominator  be  changed  from  +  to  —  or  from  — 
to  +,  the  value  of  the  fraction  will  remain  the  same. 

Thus, — =J;  and =:—ab— — a=:-\-b. 

a  — a 


Again,  f^±l'=S+c;  and  Z:Stz^=+b+c. 
a  — a 

Also,  ?L-=ff=6-c;  and  -^+'"==+I>-c. 


a  — a 

133.  Hence,  to  make  a  negative  fraction  positive  without 
altering  its  value,  change  the  sign  before  the  fraction,  and  also 
before  all  the  terms  of  the  numerator, 

rpi  a-\-b      ,  — a — b 

Thus,  — f-.=  +  — -J- ; 

c-{-d  c-{-a 

A     ■,   a—b-\-d , — a-\-b — d^ 

^  '      "SM        "^      Sabd      ' 

Also,~i?^+=:l?=-2. 
'       6  6 

CASE    II. 

IS^.  To  reduce  a  mixed  number  to  an  improper  fraction. 


SECT.  III.J^       ALGEBRAIC  FRACTIONS.  51 

RULE. 

1.  Make  all  the  fractional  parts  positive, 

2.  Multiply  the  quantity  to  which  the  fraction  is  annexed^  by 
the  denominator  of  the  fraction,  and  connect  the  product^  by  its 
proper  sign,  with  the  numerator. 

3.  Under  this  result  write  the  denominator. 

EXAMPLES. 

1.  Reduce  ^o'x-f — i^  to  an  improper  fraction. 

^ab 

^^   Sa'bx-hSa^-i-bx 
^ab 

2.  Reduce  3a+ — 21 —  to  an  improper  fraction. 

Ans.  "^-3aa:+x» 
3a"— a? 

3.  Reduce  2x+y — =L  to  an  improper  fraction. 


2x— y 


Ans. 


2a?— ^ 


4.  Reduce  a — b — — - —  to  an  improper  fraction. 


5.  Reduce  8 — b — to  an  improper  fraction. 

Ans.  5Q-6^-« 
6 

6.  Reduce  3a +9 —  ^     —  to  an  improper  fraction. 

3+a 

3+a 

CASE  m. 
135.  To  reduce  an  improper  fraction  to  a  whole   or  a 
mixed  number. 

RTTLE. 

1.  Divide  the  numerator  by  the  denominator;  the  quotient  will 
be  the  integral  part. 


52  ELEMENTS    OF    ALGEBRA.  [SECT.  III. 

2.  If  there,  he  a  remainder^  write  the  divisor  under  it,  and  con' 
necty  by  its  proper  sign,  the  fraction  so  formed  to  the  integral  part, 

EXAMPLES. 

1.  Keduce  to  a  whole  quantity. 

Ans.  5a^ — 3a^x. 

2.  Reduce  ^   ■"   ^  "*"  to  a  mixed  quantity. 

a-\-b 

Ans.  a-\-h 


a-^-h 

3    Reduce  1^^^+^^!^!=:^  to  a  mixed  quantity. 
4a6 

Ans.  c-f2a6^— ??. 
^  26 

4.  Reduce  ^^^^+^^'~"^  to  a  mixed  quantity. 

^ws.  2a +1 — — =• 
2jc* 

5.  Reduce to  a  mixed  quantity. 

ct — ax -{-or 


Ans.  a--f-aa; — — 1 -. 

ce- — aaj-f-ar 


6.  Reduce to  a  whole  quantity. 

a — 0 


Ans.(^-\-ab  +  i^, 


7.  Reduce" —  to  a  ftiixed  quantity. 

a^ — ab^ 


Ans.    a'+b'—  ^"'^^ 


^ 


8.  Reduce  ^"'^~^^^'^'     ^  to  a  mixed  quantity. 

2a^ic^ 

a^       X       aV 

CASE    IV. 

136.  To  find  the  ^eatest  common  divisor  of  two  numbers. 


In  order  to  obtain  a  general  rule  for  finding  the  greatest 
common  divisor,  we  must  observe : 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  53 

1.  If  two  numbers  are  respectively  divisible  by  a  third, 
their  sum  or  difference  will  also  be  divisible  by  the  same 
number.  Thus,  if  a  and  6  are  each  divisible  by  c,  a  4-6  and 
a — b  will  also  be  divisible  by  c;  for,  if  c  is  contained  in 
a  eight  times  and  in  b  twice,  in  a-\-b  it  will  be  contained 
ten  times,  and  in  a — b  six  times. 

2.  If  any  number  is  divisible  by  another,  every  multiple 
of  that  number  will  also  be  divisible  by  the  other.  Thus, 
if  a  is  perfectly  divisible  by  b,  2a,  3a,  4a,  or  ma  will  also 
be  divisible  by  b. 

3.  Hence,  if  t\yo  numbers  are  divisible  by  a  third,  the  dif- 
ference between  the  larger  and  any  multiple  of  the 
smaller  of  these  numbers  must  also  be  divisible  by  that 
third  number.  Thus,  if  c  is  contained  in  a  eight  times 
and  in  b  twice,  it  will  be  contained  in  3b  six  times,  and 
in  a — 3b  twice. 

4.  Also,  if  the  larger  of  two  numbers  having  a  common 
divisor  is  divided  by  the  smaller,  the  remainder  will  be 
divisible  by  the  common  divisor. 

137.  From  the  preceding  principles  we  deduce  the  follow- 
ing general  rule  for  finding  the  greatest  common  measure. 

BULB. 

1.  Divide  one  of  thk  given  numbers  by  the  other. 

2.  If  there  be  a  remainder ,  divide  the  first  divisor  by  this  re- 
mainder, 

3.  Continue  to  divide  in  the  same  manner  till  there  is  no  re- 
mainder  ;  the  last  divisor  will  be  the  greatest  common  measure. 

Jfote  1. — If,  in  the  course  of  the  reduction,  one  factor  is 
found  to  be  common  to  all  the  terms  of  one  of  the  quanti- 
ties and  not  of  the  other,  this  factor  may  be  cancelled  ;  for, 
since  only  one  of  the  numbers  is  divisible  by  it,  it  cannot  be 
a  factor  of  the  common  divisor. 

J^ote  2. — For  a  like  reason,  the  dividend  may  be  multi- 
plied by  a  factor  which  does  not  contain  a  measure  of  the 
divisor. 


64  ELEMENTS    OF   ALGEBRA.  [SECT.  III. 

EXABIPLES. 

1.  Find  the  greatest  common  divisor  of  d^ — a^x-^-daa^ — 
3af^  and  a^ — 5aa;+4a;2. 

Finst  Division, 


4>a^x —     aa^ —  3a^ 
Wx—20ax'-i-16x'' 


Dividing  by  19a;')     19ax^—19x' 
a    —     X 

Second  Division* 
a^ — 6ax-{-4>x^  \a —  x 
a^ —  ax  a — 4<a? 


— 4!ax-\-4>x^ 
— 4aa:+4a?^ 


0 
Hence,  the  greatest  common  divisor  is  a — x. 
2.  Find  the  greatest  common  divisor  of  a^ — ab^  and  c^-{- 
2ab-\-b\ 

Dividing  a^ — a¥  by  a,  we  obtain  a^ — &^. 
First  Division. 
a^+2ab+  b'  \a'-^b^ 
a'  —  b'    1 


Dividing  by  2^')    2ab-i-2b^ 
a  -\-  b 

Second  Division* 

a^—b'      \a~{-b 
a'^ab      a — b 


—ab—b^ 
—ab—b^ 


0 
Hence,  the  greatest  common  divisor  is  a-{-h. 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  55 

3.  Find  the  greatest  common  divisor  of  a* — cc*'  and  0*4- 
c^x—ax^—x^. 

First  Division, 
a* — X*  \a^-{-a^x — aa^ — a? 

o^-\-a'x — a^x^—axr^     a —   x 


— a'x+aV-l-ax* —  ap* 
— a^x — aV-|-ar''+  x* 

Dividing  by  2x^)  2aV         —2a?* 

Second  Division, 
{^-{-a'x—ax'—sc'    |a^— ^ 
fl*         — ox*  a  -f-op 

a^x         — X* 
0*0?  — x^ 


0 
Hence,  the  greatest  common  divisor  is  a^ 

4.  Find  the  greatest  common  divisor  of  a* — b*  and  o* — ft*. 

First  Division. 


a^—  b' 
a'— ah' 

\a^-lP 
a 

a  -b 
Second  Division. 

a»— a-'ft 

(y-^ab^b^ 

a'b^ab' 

ab'^y^ 
ah'— IP 

0 
Hence,  the  greatest  common  divisor  is  a — b. 


55 


ELEMENTS  OF  ALGEBRA. 


[sect.  III. 


5.  Find  the  greatest  common  measure  6(  3a'* — 6a^b-\-5a%^ 
—6ab'+2b'  and  6a'-{-8a'b—llab'-\-2b\ 

First  Division. 
Multiplying  by  2, 

3a*—  5a^64-   oa^b^—  5aZ>'+   m 


Qa*—10a'b-\-10aW—10ab^-{-  4Z>''|6a^4- Sa^b—Uab^  +  2b^ 
6a*+  Sa'b~na'b'+  2ab^  a— 3b 


—  lSa'b-^21a%^—12ab''-{-  4M 
—18a'b—24>aW+33ab^—  66* 

Dividing  by  5Z>2)    45a'^>'— 45a6^+ 106* 
9^2  __  9^j  ^.  262 

Second  Division. 
Multiplying  by  3,  6a=^+8a'6—  lla62+   26= 


18a='+24a^6—  33a62+   66^ 
ISa''— 18a'64-     4a62 


da"—  9a6+262 


2a +146 


Multiplying  by  3,  42a^6—  37a6^4-  ^^ 
126a^6— llla62+186=' 
1260^6— 126a62  4-286'* 


Dividing  by  56^) 


15a6^— 10^  N 


3a    — .  2^ 

Third  Division. 
9a*— 9a6+262|3a— 26 


.^ 


9a^— 6a6 


3a— 6 


— 3a6+262 
—  3a6+262 


0 
rience,  the  greatest  common  divisor  is  3a — 26. 

6.  Find  the  greatest  common  divisor  -of  a* — 6*  and  a* — a*6 
— 06^+6^  Jlns.  a^—l^. 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  57 

7.  Find  the  greatest  common  divisor  of  Scr* — Sa^x+as? — 
X*  and  W — 5ax-\-x^.  Ans.  a — x, 

8.  Find  the  greatest  common  divisor  oi  W — 2a' — 3a+l 
and  3a'— 2a— 1.  *^ns.  a— 1. 

9.  Find  the  greatest  common  divisor  of  0^+90^4-270 — 
98  and  o»-|-  12o— 28.  Ans,  ar-2. 

10.  Find  the  greatest  common  divisor  of  36o'5' — 18a'J* — 
27o*^>"+9o*6*    and  '21a'b^—lSa*b*—9a'lf'. 

Ans.  9o*^>"— 9o'6«. 

138.  The  greatest  common  divisor  of  more,  than  two  num- 
bers may  be  obtained  by  finding,  in  the  first  place,  the 
greatest  common  divisor  of  two  of  them,  and  then  of  that 
divisor  and  the  third,  and  so  on.  The  last  divisor  thus 
found  will  be  the  greatest  common  divisor  of  all  the  quan- 
tities. 

EXAJIPLE. 

Find  the  greatest  common  divisor  of  o* — J*,  a'+2a'J4-2a6^ 
+  &»,  and  a*^a^b'+b\ 

First  Division, 
o'+2o*&-f  2a^»»+  6'    •  |o^— y 
a"  —  &»       1 


2J).     2o'6+2aA«-f2ft* 
a*  -\-  ab-{-  l^ 
Second  Division, 

o'+a'J+o^         o—  6 


t^b—ab^—h' 


0 
Hence,  the  greatest  common  divisor  of  the  first  two  num- 
bers is  a*-{-ab-\-b^, 

H 


58  ELEMENTS    OF   ALGEBRA.  [sECT.  HI. 

Third  Division. 


—a^b+¥ 
—a^b—a'b^—aW 

aW+aW-\-b'' 
a^^J^ab^-^b^ 

0 
Hence,  the  greatest  common  divisor  of  the  three  num- 
bers is  a^-\-ab-\-b^. 

CASE    V. 

139.  To  reduce  a  fraction  to  its  lowest  terms. 

RULE. 

Divide  the  two  terms  of  the  fraction  by  their  greatest  common 
divisor. 

JsTote. — To  show  that  the  value  of  the  fraction  will  not  be 
altered  by  the  operation  indicated  in  the  preceding  rule,  we 
will  demonstjate  the  following  theorem : 

Theor.  If  both  terms  of  a  fraction  be  divided  by  the  same 
quantity^  its  value  will  not  be  altered. 

Let  aim  and  am  represent  the  numerator  and  denominator 
of  an  algebraic  fraction  of  any  assignable  value,  m  repre- 
senting any  whole  or  fractional  number  whatever : 

Then  the  fraction  ^!L^—dbm^am'—b. 
am 

Dividing  both  terms  of  the  fraction  by  the  indefinite  num- 
ber w,  and  reducing  — —ab-^a—h, 
a 

Hence  (by  Ax.  2),  ^--=.—  ',  which  was  to  be  demon- 
am       a 

strated. 

EXAMPLES. 

1.  Reduce  _- —  to  its  lowest  terms. 

Viiab 


SECT.  III.l  ALGEBRAIC    FRACTIONS.  59 

■*  • 

The  greatest  common   divisor  is   6a;  hence,       ^^ 


\%ab 


-———1-^=-—,  which  is  the  simplest  fdrm  of  the  fraction. 
18aZ>~6a     36  ^ 

2.  Reduce  ill^  to  its  lowest  terms.  *dns.  ?^ 

2\aa*  3 

3.  Reduce  ''"^^^'-'*°'^'f +^'''^^  to  its  lowest  terms. 
The  greatest  common  divisor  is  7a'6V.  Ans.  _^Z1  Jl_. 

4>.  Keduce  - — __  to  its  lowest  terms. 


Of, 5 


The  greatest  common  divisor  is  9aa:.      Arts,  ^ 


5.  Reduce  5«^^^4- lOa^ar^  ^^  j^^  1^^^^^  ^^^^^^ 

aV+2aV 

The  greatest  common  divisor  is  a*a?*+2aj:*.     Atis.  — . 

a 

«' 1?  ^,««^*— Sax'— 8aV4- 180*0?— 8aV    .     , 

6.  Reduce — —L to  its  lowest  terms. 

ar*— aj^— 8a«x-f-6a» 

The  greatest  common  divisor  is  3?-\-^lax — 2a'. 


X — 3a 

to  its  loTvi^st  tftrms. 

2i'+3a^>-}-a» 


p»    Ti    3         2a&' — a'J — a*  ,      .    i 

7.  Reduce  — — — to  its  lowest  terms. 


Am,  ±Z±, 
6+a 

b.  Reduce  ! to  its  lowest  terms. 

6aj7 — 8a 

2 

9.  Reduce  — ^—  to  its  lowest  terms.  Arts,  ^        . 

a*+i*  1 

10.  Reduce  ^^+2aV+2a^+a:^  ^^  .^^  j^^^^^  ^^^^^ 

5a*+5a'a? 


60  ELEMENTS  0^  ALGEBRA.  [sECT.  III. 

CASE    VI. 

140.  To  find  the  least  common  multiple  of  two  or  more 
numbers. 


The  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  which  can  be  divided  by  each  of  them  without 
a  remainder.  The  reason  for  the  foUoAving  rule  will  be  suf- 
ficiently obvious  without  farther  illustration. 

RULE. 

1.  Resolve  the  numbers  into  their  prime  factors. 

2.  Select  all  the  different  factors  which  occur ^  observing^  when 
the  same  factor  has  different  powers,  to  take  the  highest  power. 

3.  Multiply  together  the  factors  thus  selected,  and  their  pro- 
duct will  be  the  least  common  multiple. 

\  EXAMPLES. 

1.  Find  the  least  common  multiple  of  Sa^cc^y,  l^a^b^x^  and 
IQa^'^cx. 

Resolving  them  into  their  prime  factors, 
8a Vy = 2^*  X  a"  X  0?^  X  y 
*  na%'x=2'xa'xx         xb^X^ 

16aWcx=2'Xa'xx        xb^XC 
The  different  factors  are  2",  a*,  x^,  y,  Z>',  3,  and  c. 
Hence,  the  least  common  multiple  is  2''x3xa''x2>*xcx 
x^'Xy—4fSa'^b^cx'^y. 

2.  Find  the  least  common  multiple  of  12a^J^,  IGa'^bc^,  and 
24a.  Jlns.  48a^JV. 

3.  Find  the  least  common  multiple  of  Sa^b,  5a,  7a^c,  12a', 
15a^  ISa^Jc,  and  35a*Z'V.  Jlns.  1260a«JV. 

4.  Find  the  least  common  multiple  of  12a''y+  12a^by,  Gce^i^-^ 
na'bf^-\-6abY,  and  4ay. 

Resolving  the  numbers  into  their  prime  factors, 
12a=y4-12a%  =12aya+Z»)=2'x3xa=^xy  X(a-\-h) 


6ay  +  l^a'hy^+GabY  =  Qay\a^-\- 
%ib-^¥) 

4aY  =2'X        a^xy' 

The  different  factors  are  2^  3,  a^,  /,  and  {a+bf. 


t=2  x3xa  xy^X(a+hY 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  6l 

Hence,  the  least  common  multiple  is  2*x3xa^Xy'x(a-h 
bY=l'2aY{a  +  by. 

5.  Find  the  least  common  multiple  of  a* — J*,  a+J,  and 

6.  Find  the  least  common  multiple  of  a-f  ^>  ^ — ^>  or-\-ah-\- 
^,  and  a»-  ah-\-h^,  Ans,  a^—b\ 

CASE   VII. 

141.  To  reduce  fractions  to  equivalent  ones  having  a  com- 
mon denominator. 

RULE. 

1.  Multiply  each  numerator  into  all  the  denominators^  except 
its  own^  for  the  new  numerators. 

2.  Multiply  all  the  denominators  together  for  the  common  cfe- 
nominator.  ^ 

J^ote  1. — It  will  be  perceived  that,  by  the  operations  indi- 
cated in  the  preceding  rule,  the  terms  of  each  fraction  are, 
in  effect,  multiplied  by  the  product  of  the  othfer  denominators, 
i.  e.,  the  numerator  and  denominator  of  each  fraction  are 
multiplied  by  the  same  number.  To  show  that  the  value  of 
the  fractions  is  not  altered  by  this  transformation,  it  is  only 
necessary  to  demonstrate  the  following  theorem  : 

Theor.  If  both  terms  of  a  fraction  be  multiplied  by  the  same 
number^  the  value  will  not  be  altered. 

Let  ab  and  a  represent  the  numerator  and  denominator 
of  an  algebraic  fraction  of  any  assignable  quantity  : 

Then  the  fraction  —=iab-^a=zb. 
a 

Let  m  represent  any  whole  or  fractional  number  what- 
ever ;  then  multiplying  the  terms  of  the  fraction  by  fw,  we 

have  ^ — =abm-7-am=b. 
am 

Hence,  since  ^=b   and  ^=i   (by  Ax.  2),  ^=^, 
a  am  a      am 

which  was  to  be  demonstrated.  ' 

Jfote  2.~Mixed  numbers  should  be  reduced  to  improper 

6 


62  ELEMENTS    OF    ALGEBRA.  [SECT.  III. 

fractions,  and'all  the  fractions  should  be  made  positive  be- 
fore they  are  reduced  to  a  common  denominator. 

JVb^e  3. — Whole  numbers  or  integers  can  be  put  under 
the  form  of  a  fraction  by  writing  1  for  a  denominator  un- 
der them,  and  then  be  reduced  to  a  common  denominator 
with  fractions. 

EXAMPLES. 

1.  Reduce  -,  -,  and  —  to  equivalent  fractions,  having  a 

b  d  y        ' 

common  denominator. 

axdxy—ady^  first  numerator. 
cxlxy—hcy^  second  numerator. 
xxhxd—hdx,  third  numerator. 
bxdxy=^hdy,  the  common  denominator. 

Hence,  the  values  of  the  fractions  are  —^^  — ^,  and  — . 

bdy  bdy  bdy 

2.  Reduce  — ,  - — ,  and  —  to  equivalent  fractions,  having  a 

common  denominator,    ^ns. i,  ,  and ^. 

l^cxy  llcxy  \2cxy 

3.  Reduce  — ,  — ,  and  __  to  equivalent  fractions  having  a 

ax  36  7c? 

common  denominator. 

a         SUd      Ua'dx         ,    Ibabx 

Jins. , ,  and . 

2labdx    ^labdx  21abdx 

4},  Reduce  —  and  to  equivalent  fractions  having  a 

6b  3c 

common  denominator.  ,dns. and it — . 

15k  156c 

5.  Reduce  -,  — I^,  and  a  (or  -)  to  equivalent  fractions 

b    c-\-d  1 

having  a  common  denominator. 

^       ac-{-ad   3bx — 26        .  abc-\-abd 
bc-\-bd^   bc-\-bd  '  bc-\-bd 

6.  Reduce     ^      ,  -,  and  -— J^  to   equivalent  fractions 

4a;       5  c-\-d 

having  a  common  denominator. 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  63 

J,       25«c4-25af/— 5c— 5rf   12cxH-12dir        ,  20&x  +  4aj?y 
^  20cx-f-20(^        '  20cx-f20(/a?'  20cx+20^* 

7.  Reduce  — -1-  and     "^    to  equivalent  fractions  having 

a  3 

a  common  denominator.  Ans,  — -—  and  — — — . 

3a  3a 

8.  Reduce  ,  — -,  and  —  to  equivalent  fractions  hav- 

76a?     a  — 5 

ing  a  common  denominator. 

a  25  ad       —I05bcx         ,     2Sbdx 


—^bbdx"  —Sbbdx'  —Sbbdx 

2     I     IL2  O IL  Z?     2 

9.  Reduce  — l!l-, ,  and  — to  equivalent  frac- 

2a      a — b  2aH-2a6 

tions  having  a  common  denominator. 

Ans  ^'— ^^^'    12^-fl2a^^  and  1?^— ?^ 

3 -r^ O  Oj2       y  _j_  4 

10.  Reduce  and  i—  to  equivalent  fractions 

4a  a-|-a? 

having  a  common  denominator. 

jj^  3ax^+3x'—2a—2x  ^^^  Soa?^— 4.ax+ 16a 
4a'+4ax  4a*-f4aa; 

142.  To  reduce  fractions  to  their  least  common  denomi- 
nator. 

SITLB. 

1.  Find  the  least  common  multiple  of  all  the  denominators  of 
the  given  fractions,  and  it  will  be  the  least  common  denominator, 

2.  Divide  the  least  common  denominator  by  the  denominator 
of  each  fraction  separately,  and  multiply  the  quotient  by  the  re- 
spective numerators,  and  the  products  will  be  the  numerators  of 
the  fractions  required, 

EXAMPLES. 

1.  Reduce  — -  and  — — -  to  their  least  common  denomi- 

Sjt  4aV 

nators. 

ar«    =2»xa« 
4flV=2'xa?»xaP 


64  ELEMENTS    OF   ALGEBRA.  [sECT.  III. 

Hence,  the  least  common  multiple  is  2^x07^X0^=80^. 
Then,^^xSa'=:a'xxSa'z=z  Sa'x 


?.'^'x5a6=2    X6ah=:10ai 
4aV 


)■  new  numerators. 


^;,..^andJ^. 

8aV  Sa'x' 

2.  Reduce  — -,  Jl,  and to  their  least  common  de- 

Qoric    ^cLcrxu  Sa?^ 

nommators.  Ans.  y^^r-.->  tAi  and  — _-.. 

8acV'   8acV  8acV 

3,  Reduce  _ -, ,  and to  their  least  common 

a^ — or   4a — 4a?  a-\-x 

denominator. 

Ans.  _i-^_,  Mt^,  and  ?0^^^=20^.  - 


CASE    VIII. 

ADDITION  OF  FRACTIONS. 

143.  Theor.  If  two  fractions  have  a  common  denominator^  their 
sum  will  he  equal  to  the  sum  of  their  numerators-divided  by  the 
common  denominator. 

Let  —  and  — ,  represent  two  fractions  whose  common  de- 
a  a 

nominator  is  a  ; 

rpr        am  .an am-^an  , 

^    a       a  a 

For,  '^=m,  and  2^=»;  therefore,  ^+^=m+». 
a  a  a       a 

But,  ^'^~^^'^={am-^an)^a=m-^n  ; 
a 

Hence  (by  Ax.  2),  — +— =^^"'"°^,  which  was  to  be  de- 
ex       a  a 

monstrated. 

144.  Hence,  to  add  fractions,  we  obtain  the  followingf 

general 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  65 

RULE. 

1.  Reduce  the  fractions  to  equivalerit  ones,  having  a  common 
denominator ^  and  make  them  all  positive. 

2.  Jldd  all  the  numerators  together,  and  under  their  sum  write 
the  common  denominator. 

3.  Reduce  the  resulting  fraction  to  its  lowest  terms, 

EXAMPLES. 

1.  Add  toffether  — ,  — ,  and  — • 

^  2A'y         la 

Reducing  the  fractions  to  a  common  denominator^ 

3^     2a     3b_l06a'     2Sa'b      30Z>' 
b      ~b      la~10ab     lOab      TOab 
Adding  the  numerators  of  the  reduced  fractions, 

lOba'    2Sa'b     30b'  _l05a'-\-2Sa'b-{-30b'      ^ 
lOablOab     lOab  lO^b  ^' 

2.  Add  together  ±  and  ?5±^.  ^^ns.  ^^±^b±Uc^^hx^    * 

2b  a\-b  2ab-\-2l^ 

3.  Add  together  ?^+ \  ^^  and  1     Jlns.  1^±IL 

^  3     '      5     '         7  105 

4.  Add  together  ^±?f ,  tl^',  and  -t 

b3^-\^xy. 

5.  Add  together  ^±^  and  ?IZ*.  ^ns.  ^'+^. 

a—b         0  +  4  a'— 4* 

6.  Add  together  ?,  ^i^,  and  ±±=f?. 

6      cd  bed 

Jlns  ?!£^.^±i 

7.  Add  together  1^^  and  ^.  Ans.  ?^^. 

8.  Add  together  -^  and  —,  jlns.  °'+^'. 

a-f6  a — b  a^— ^ 

9.  Add  together  _?1,  —?,  and  —1 

^«j   5g^— 3/>>'— 4qx— 4^g 
4Mb+W. 
I 


66  ELEMENTS    OF   ALGEBRA.  [sECT.  III. 

10.  Add  together  -,  — ^2^  ,  and  — ?^. 
a       b — 1  fe+1 


ab^ 


11.  Add  together  2a+^  and  4^+?^. 
5  4 


•dns,  6a + — '■ . 

20 

12.  Add  together  a — —  and  b-\- . 

be 

a  .  7  .  2abx — Sea?* 

be 

CASE    IX. 

SUBTRACTION  OF  FRACTIONS. 

145.  Theor.  If  two  fractions  have  a  common  denominator  f  their 
difference  is  equal  to  the  numerators  divided  by  the  common  de- 
nominator. 

Let  —  and  —  represent  two  fractions  whose  common  de- 
a  a 

nominator  is  a  ; 

rpi         am     an     am — an 

ihen,  — — — = . 

a       a  a 

T^      am  J  an  .i       f        am     an 

ror  — =m^  and  — —n  j  therefore,  — — — =m — n. 
a  a  a       a 

■D   ,    am — an     /  \  • 

But, = (am — an) —a— m — n, 

a 

Hence  (by  Ax.  2),  — — — =^       ^^,  which  was  to  be  de- 
a       a  a 

monstrated. 

146.  Hence,  to  subtract  one  fraction  from  another,  we 
obtain  the  following  general 

RULE. 

1.  Reduce  the  fractions  to  equivalent  ones,  having  a  common 
denominator^  and  make  them  all  positive. 

2.  Subtract  the  numerator  of  the  fraction  to  be  subtracted  from 
that  of  the  other  fraction^  and  under  their  diffei'ence  write  the 
common  denominator. 


SECT."  III.]  ALGEBRAIC    FRACTIONS.  67 


EXAMPLES. 

1.  From  ??  subtract  if. 
3b  bd 

Reducing  the  fractions  to  a  common  denominator,  dec. 

2a_^c_l0td_12bc_10ad—12bc    ^^^ 

3b     5d~lbbd     lbbd~       15W 

%  From  ^J±  subtract  1  Jlns,  ^^"^^"^^ 

c  y  cy 

3.  From  ^1  subtract  e!^±^. 

lb  3ax 

76         3aJ7    ~21aAj?        21aZ»a7      "~  ^labx 

4.  From  ^  subtract  ?i±i.  wf;w.  ^^-^^^ 


5                    x+\  bx+b 

5.  From subtract .  Ans.       ^ 


x—y  x-\-y  x'—f 

6.  From  ^±?  subtract  -J_.  vfn*.  ?JZ±Z% 

y  x^ — 2  a?*y — 2y 

7.  rrom subtract -.  Ans. 


ax — x^                 ax-\-3^  a^—a* 

8.  From  6a-[.—  subtract  Sa-\-—,  Am.  2a+^^^~^. 

0?                           c  ex 

9.  From  6ot— 1^!±1  subtract  f7»4-?.  ^^.  5m— ?2^il. 

2                           5  10 

10.  From .  subtract 


.^;w. 


8a*— 2a*6^+4a'6»— a6«     6*— 4a« 


CASE    X. 

MULTIPLICATION  OF  FRACTIONS. 
147.  Theor.  Tht  'product  of  two  fractions  is  equivalent  to  the 
product  of  the  numerators  divided  by  the  product  of  the  denomi- 
nators. 

Let  -  and  _  represent  any  two  fractions : 

Then  will  ^X^=^'. 
b    b'    bb'' 


68  ELEMENTS    OF   ALGEBRA.  [sECT.  III. 

For,  letting  v  represent  the  value  of  tjie  first  fraction,  and 

v'  the  value  of  the  second,  we  shall  have  -=v,  and  -  =iV'. 
'  b  b' 

Multiplying  the  two  equalities  together  (Ax.  5),  -  X  -  =v.  V. 

b     b' 

Multiplying  the  equality  -=v  by  b  (Ax.  5),  a=bv, 
b 

a' 
Multiplying  the  equality  -=iv'  hy  b'  (Ax.  5),  a'—b't/. 

Multiplying  the  last  two  equalities  together  (Ax.  5),  aa'=z 


bv  .b'v'=bb'  xvv'. 

lualitv  bv  bb'  (Ax.  G\  

bb' 


Dividing  the  last  equality  by  bb'  (Ax.  6),  — =vv\ 


Hence  (by  Ax.  2),  _x  -=^,  which  was  to  be  demonstra- 
ted. 

CoROL.  The  product  of  any  number  of  fractions  is  equiva- 
lent to  the  product  of  the  numerators  divided  by  the  product 
of  the  denominators : 

ihus,  -X  — X — = ► 

'  b     b'     b"     bb'b'^ 

148.  From  the  foregoing  theorem  we  infer  the  following 
general  rule  for  the  multiplication  of  fractions. 

*i'       '     "         RULE. 

1.  Multiply  the  numerators  together  for  a  new  numerator,  and 
the  denominators  together  for  a  new  denominator. 

2.  Reduce  the  resulting  fraction  to  its  lowest  terms. 

EXAMPLES. 

1    i\/r  u-  1     3^2,      bah  n^^   ^a\bah     Ibah 

1.  Multiply  —  bv .  ^ns.  — -  x -—-=-——,. 

^  ^  46     ^  Ib'd  46     Wd    2Sb'd 

2.  Multiply  ?^±^  by  ^. 

^  ^     Sax      ^  Sdx^ 

^^^   3a'+6     2ac^_6ah+2abc^_3a''c-\-bc^ 


Sax       3dx'         24>adx'  ndx' 

3a^ 


3.  Muhiply  — L—  by Jins. -!— . 

^  ^    a—1      ^  a+1  a'—l 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  69 

'  4.  Multiply  ?^^  by  -J^-.  Jim.  ^^ 

5.  Multiply by  — — -.  Ans.  1. 

6.  Multiply  ±ZI  by  ^!±f?.  wf^.    ^'"^ 


ex  c-\-x  c^x-i-ca^ 

7.  Multiply  —,  ^,  -,  and  _i_  together. 

m      y      c         n — 1 

Ans.  ^i''-^^ 
cmny — cmy 

8.  Multiply  —,  — .  and  ^  together.  jJns.  A. 

9.  Multiply by  — - — -.  Ans.  -- -L- — . 


10.  Multiply  e!^^^  by -Jl—      ^  Jlns.?^±^. 

J^ote  1. — Since  every  integer  can  be  expressed  in  the  form 
of  a  fraction  by  writing  1  under  it  for  a  denominator,  it  is  evi- 
dent that  an  integer,  or  whole  number,  may  be  multiplied 
into  a  fraction  by  multiplying  the  numerator  of  the  fraction 
by  the  whole  number,  while  the  denominator  remains  the 

same. 

mi  ^b     a^  h     ab 

Thus,  aX_=_x-= — . 
c     1     c      c 

^ote  2. — If  the  denominator  is  divisible  by  a  whole  num- 
ber, dividing  the  denominator  multiplies  the  fraction. 

Ihus,  _xc=-_^=_;  for,  _xc=_-X-=— =-. 
be  bc—e     b  be  be     1     be     b 

CASE    XI. 

DIVISION  OF  FRACTIONS. 
149.  Theor.  If  one  fraction  be  divided  by  another  fraetion, 
the  quotient  will  be  equivalent  to  the  product  of  the  fracticmal 
dividend  multiplied  by  the  fractional  divisor  inverted. 

Let  -  and  ~  represent  any  two  fractions : 
0         b' 

Then  will  ?-?.'=?*:: 

b    b'    ab 


70  ELEMENTS    OF    ALGEBRA.  [SECT.  III. 

For,  letting  v  represent  the  value  of-,  and  v'  the  value  of  - ; 

b  h' 

Then,  -=v  and  -=:v':  and  -^-=v-^v'. 
b  b'        '  b  '  b' 

Multiplying  the  equality  -—V  by  bb'  (Ax.  5),  ab'=bb'v.  , 
b 

Multiplying  the  equality  -^=:v'  by  bb'  (Ax.  5),  a'b—bb'v'. 

Dividing  the  former  by  the  latter  of  the  last  two  equalities, 

ab'      bb'v 


ba'     bb'v' 


:V-T-V'. 


Hence  (by  Ax.  2),  — 1-_=__,  which  was  to  bedemonstra- 

b     b'     ba' 

ted. 

150.  From  the  foregoing  theorem  we  infer  the  following 
general  rule  for  the  division  of  one  fraction  by  another. 

RULE. 

1.  Invert  the  fractional  divisor, 

2.  Then  proceed  as  in  multiplication j  and  the  product  thus 
found  will  be  the  quotient  required, 

EXAMPLES. 

1.  Divide  ?^  by  ^.• 

46     ^  6d' 

4A"  ■  Q^~Tb      5c'    'WJI~TOb?' 
2.Divide?^!±l^by5^. 

3a^-{-2b_^3a_3a'  +  2b ^5b_15a''b-^10b''  ^^^ 
'WTc   '  56      26+c~     3a       6ab-f3ac~' 

3.  Divide  -^  by  -.  Jlns.  —, 

1—a    ^  5  1—a 

4.  Divide  ^-±tl  by  1-,  '  Ans,  _?^1__. 

e—y^     ^  c—b  c'-\-bc-hb' 

5.  Divide  -^^-Z^  by  «!±^l        Ans.  ^±l=a^t 

a'—2ab-{-b'     ^    a—b  a  a 

a    r\'   'A     3a — 3b  ,      5a  — 5b  n       3 

6.  Divide bv  .  Jins.  -. 

a-\.d      ^     a+d  5 


SECT.  III.]  ALGEBRAIC    FRACTIONS.  71 

7.  Divide  -^  by  -Ij.  Ant.  _,    '^      .. 

8.  Divide  5^  by  ^-III.  ^n..  1?^. 

9.  Divide  ^  by  5^1^  ^...  ^.. 

3c*      ^         7  •    6c» 

10.  Divide     2aV-2c^     .     g'+ac+c^      ^;*,.  2(a3+c»). 

JVo^e  1. — When  a  fraction  is  to  be  divided  by  a  whole 
number,  or  a  whole  number  by  a  fraction,  write  the  whole 
number  in  the  form  of  a  fraction  by  making  its  denominator 
1,  and  then  proceed  as  before. 

Thus  «-c=«-^=^xl=f: 
0  b     1     0     c     be 

A«j  b     a.b_a^c_ac 

And,  a--=   -_=    x-=--. 
c     1     c     1     ^      b 

J^ote  2. — The  reciprocal  of  a  fraction  is  expressed  by  the 

fraction  inverted : 

Thus,  the  reciprocal  of  J  is  -,  or  1^  ^: 
o       I  b 

But,  l-r-f!=lx-=-;  hence,  the  reciprocal  of  -  is  _. 
baa  b      a 

^ote  3. — If  the  numerator  or  denominator  of  a  fraction 
has  a  rational  coefficient,  the  expression  may  be  reduced  to 
a  simpler  form,  on  the  principle  that  multiplying  the  denomi- 
nator of  a  fraction  has  the  same  effect  upon  its  value  as  di- 
viding the  numerator ;  and  multiplying  the  numerator  has 
the  same  effect  as  dividing  the  denominator. 

Thus,    l.¥=|x«=|. 

3a-h2A     3a+26     ^      3a+2^       24a-hl66' 
q     f_^^^__^     d_ad 

'  yi'  d'~i^c~'Tc 


72  '  ELEMENTS  OF  ALGEBRA.  [sECT.   IV. 


SECTION  IV. 

OF   EaUATIONS. 

151.  An  EQUATION  is  the  algebraic  expression  of  two  equal 
quantities  connected  by  the  sign  of  equality. 

152.  The  monomial  or  polynomial  quantity  which  is  writ- 
ten on  the  left  of  the  sign  of  equality  is  called  the  first  mem' 
ber  ;  that  which  is  written  on  the  right,  the  second  member. 

An  equation,  then,  is  composed  of  two  members  j  and  each 
member  is  composed  of  one  or  more  terms. 

153.  The  two  members  of  the  equation  must  be  composed 
of  quantities  of  the  same  kind ;  that  is,  dollars  must  be  put 
equal  to  dollars,  weight  equal  to  weight,  &c. 

154.  Equations  are  distinguished  into  different  degrees,  ac- 
cording to  the  highest  power  of  the  unknown  quantity.  If  it 
involve  only  the  first  power  of  the  unknown  quantity,  it  is 
called  an  equation  of  the  first  degree.  If  the  highest  power 
of  the  unknown  quantity  be  the  second  power,  it  is  called  an 
equation  of  the  second  degree  ;  if  it  be  the  third  power,  an 
equation  of  the  third  degree,  &c. 

Thus,  x=a  is  an  equation  of  the  first  degree. 

x^=za  ) 
,       _    ^  are  equations  of  the  second  degree. 

o^-\-x^—a>  \  are  equations  of  the  third  degree. 
x^-\-3:?-\-x  —  aj 

155.  The  solution  of  a  problem  is  the  method  of  discover- 
ing, by  analysis,  the  value  of  the  unknown  quantity  involved 
in  the  conditions  of  the  problem,  and  consists  of  two  parts. 

1.  The  translation  of  the  problem  from  common  into  algebraic 
language  ;  or  the  expression  of  its  conditions  in  the  form  of  an 
equation  by  means  of  algehraic  symbols. 

2.  The  reduction  of  the  equation  to  such  a  form  that  the  un- 


SECT.  IV.]  EQUATIONS.  73 

known  quantity  may  stand  ly  itself^  and  form  one  member  of  the 
equation^  while  the  known  quantities  form  the  other. 

156:  No  general  rule  can  be  given  for  translating  the 
problem  from  common  into  algebraic  language ;  only  that 
the  algebraic  expression  shall  exhibit  the  same  relations,  and 
indicate  the  same  operations  as  those  implied  in  the  original 
statement  of  the  problem. 

157.  Proportions  may  be  converted  into  equations  by  taking 
the  product  of  the  first  and  fourth  terms  for  one  member, 
and  the  product  of  the  second  and  third  for  the  other.  Thus, 
if  a :  6 : :  c :  </,  converting  the  proportion  into  an  equation,  we 
shall  have  a x d=  bxc ;  or,  if  2 : 4 : :  8 :  16,  we  shall  have  2 x 
16=4x8. 

158  The  reduction  of  an  equation  involves  the  following 
general  axiom :  If  equal  operations  be  performed  upon  equal 
quantities,  the  results  will  be  equal.     Hence, 

1.  If  equal  quantities  be  added  to  both  members  of  an  equation^ 
the  equality  of  the  members  will  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  from  both  members  of  an 
equation,  the  equality  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  the  same 
number,  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same  num- 
ber^  the  equality  will  not  be  destroyed. 

5.  If  both  members  of  an  equation  be  involved  to  equal  powers y 
the  equality  will  not  be  destroyed. 

6.  If  equal  roots  of  both  members  of  an  equation  be  taken^  the 
equality  will  not  be  destroyed. 

159.  The  verification  of  a  problem  consists  in  substituting 
the  value  of  the  unknown  quantity  for  the  unknown  quantity 
itself  in  the  given  equation,  and  thereby  ascertaining  whether 
it  answers  the  conditions  of  the  problem. 

160.  Equations  are  either  numerical  or  literal.  Numerical 
equations  contain  numbers  only,  excepting  the  unknown 
quantity.  In  literal  equations,  the  given  quantities  are  repre- 
sented by  letters. 

7  K  ♦ 


Let  it  be  required  to  , 


74  ELEMENTS    OF   ALGEBRA.  [sECT.  IV. 

EaUATIONS  OF  THE  FIRST  DEGREE,  INVOLVING  ONE 
UNKN0V7N  aUANTITY. 

161.  There  may  be  three  cases  of  equations  of  this  nature, 
viz. :  When  the  known  and  unknown  quantities  are  con- 
nected by  addition  or  subtraction^  by  division^  or  by  multipli- 
cation. 

CASE    I. 

162.  In  this  case  the  unknown  quantity  is  connected  to 
known  quantities  by  addition  or  subtraction. 

Ito) 
reduce  the  equation  ) 

Adding  32  to  both     )         9x-32+32=86+32+&.. 
members  . ) 

Subtractinff  8a;  from  ) 

,    ,,  \  f9a?— 32+32— 8a;=86  +  32+8a;— 8a?. 

both  members  \ 

Cancelling        -         -  9ir— 8x^=86  +  32. 

Reducing  -         -  a;=118. 

163.  These  operations  will  suggest  the  following  general 
rule  when  the  unknown  and  known  quantities  are  connected 
by  the  signs  plus  or  minus. 

RULE. 

1.  Transpose^  so  that  all  the  unknown  quantities  may  he  in  the 
first^  and  the  known  in  the  second  member  of  the  equation  ;  ob- 
serving to  affect  the  terms  transposed  with  the  contrary  sign. 

2.  Reduce  each  member  to  a  monomial 

EXAMPLES. 

1.  Reduce  the  equation  6x — 5+3?=  12 — x-\-lx. 

Transposing        -         6x-\-x-\-x — 1x^12-^5. 
Reducing  -  x=ll. 

%  Reduce   the    equation   14— 8a:+5  =  3a7+28+6a;— 2a7— 
16a:.  ^ns.  a;=9. 

3.  Reduce  the  equation  x — 12+7a?— 8x=:4— a:+20. 

^ns.  a:=36. 

4.  Reduce  the  equation  — 6a;— 32+1  Ox =84+ 3x— 100. 

Ans.  a;=16. 


8BCT.  IV.]       EQUATIONS    OF    THE    FIRST    DEGREE.  75 

5.  Reduce  the  equation  20 — 18a?-f-4.4.H-x=70— ISx— 5. 

Ans.  xi=l. 

6.  Reduce  the  equation  4x+25-|-3ar=6x-f  80. 

Am.  a: =55. 
CASS  n. 

164.  In  this  case  the  unknown  and  known  quantities  are 
combined  by  division. 

Let  it  be  required  to  reduce  the  eqilation  — — -=8. 

Multiplying  both  members  of  the  equation  by  4,  the  least 
common  multiple  of  the  denominators,  the  equation  be- 

comes — — — =32. 

4        2 

Reducing  the  fractions  to  whole  numbers,     3x — 2a:=32. 
Reducing  the  terms       -        -         -        -  a:=32. 

165.  Hence,  to  free  an  equation  of  fractions,  we  have  the 
following  general 

RULE. 

1.  Multiply  both  members  of  the  equation  by  the  least  common 
multiple  of  the  denominators. 

2.  Reduce  the  improper  fractions  thus  produced  to  whole  num- 
bers. 

3.  Transpose  and  reduce  the  terms  as  before. 

Kote  1. — Instead  of  finding  the  least  common  multiple  of 
the  denominators,  the  equation  may  be  multiplied  by  each 
denominator  successively. 

Kote  2. — When  a  minus  fraction  is  cleared  from  its  denomina- 
tor, the  sign  before  each  term  of  the  numerator  must  be 
changed. 

EXAMPLES. 

1.  Reduce  the  equation  1^— ?^-|-2=8. 
5       4 

The  least  common  multiple  of  4  and  5  is  20  ;  multiplying 

both  members  by  this,  the  equation  becomes 

5        4 


76  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

Reducing  the  fractions  to  whole  numbers, 

16a?-15x+40=160. 
Transposing        -        -        -        .       iQx — 15cT=160 — 4iO. 
Reducing ir=120. 

2.  Reduce  the  equation  ??—??+?=  11. 
^  3       4      6 

Multiplying  both  members  by  3,  2x—^-\--=33. 

Multiplying  by  4    -         -         -     Sx—9x-\-2xz=zl32. 
Reducing       -        -        -   .     -  a;=132. 

CASE    III. 

166.  In  this  case  the  kno\X^n  and  unknown  quantities  are 
combined  by  multiplication. 

Let  it  be  required  to  reduce  the  equation  — +-=17. 

5      4 

Clearing  of  fractions         -        -        -       12a?-|-5T=340. 

Reducing  the  terms  -         -         -  17ir=:34<0. 

Dividing  the  equation  by  17     -         -  37=20. 

167.  Hence,  if  the  unknown  quantity,  after  the,  equation 
has  been  cleared  of  fractions  and  the  terms  reduced,  has  a 
coefficient,  the  reduction  may  be  completed  by  the  following 

RULE. 

Divide  both  members  of  the  equation  by  the  coefficient  of  the 
unknown  quantity. 

EXAMPLES. 

1.  Reduce  the  equation  13ic+31=&r-|-76. 
Transposing     -         -        -        -         13^— 8a?=76— 31. 
Reducing  -         -         -         -  5a;=:45. 
Dividing  by  coefficient  of  a?       -  x=9, 

Q/M     5ii?     3x 

2.  Reduce  the  equation  — + — — — =5. 

4       3       5 

Clearing  of  fractions  45a?-|-100a; — 36a?=300. 

Reducing  -         -         -  1090,-300. 

•n-  'A-                                                                 300     o82 
Dividmff  -        -        -  X— — —Q, 

^  109       109 


SECT.  IV.]       EQUATIONS  OF    THE    FIRST    DEGREE.  77 

3.  Reduce  {he  equation  ^+^+-=12.     ^ns.  a:=:ll^. 
4*.  Reduce  the  equation  12x+_— 1  =  16.         ^ns.  Ify, 

At 

5.  Reduce  the  equation  _— 5  =  ~^_     Am.  a? =20. 

4  3 

6.  Reduce  the  equation  ^=.?-|-^+10.     Am,  «=27A. 

5  2     3 

7.  Reduce  the  equation  a?-f— -+^=26.        Ans.  x—VL 

8.  Reduce  the  equation  j:4--+— =81.        Am,  a?=36. 

2     4> 

9.  Reduce  the  equation  a;+a?4-^=100--2?. 

Am.  x=39. 

10.  Reduce  the  equation  x+?4-^+^+4-=14'6- 

2     4      7      14 

.^;i5.  a: =56. 
168.  Combining  the  principles  discussed  in  the  preceding 
three  cases,  we  have,  for  the  solution  of  all  equations  of  the 
first  degree  involving  only  one  unknown  quantity,  the  fol- 
lowing general 

RULE. 

1.  Char  the  equation  af  fractions. 

2.  Transpose  the  terms^  so  as  to  bring  all  the  unknown  quanti- 
ties into  the  first^  and  the  known  into  the  second  member  of  the 
equation. 

3.  Reduce  each  member  to  a  monomial. 

4  Divide  the  equation  by  the  coefficient  of  the  unknown  quan- 
tity. 

JVote  1. — If  the  unknown  quantity  in  the  result  is  negative, 
change  the  signs  of  all  the  terms  in  the  equation. 

Thus  -        -     4a^-5x=9— 12. 

Reducing       -         -        —  x= — 3.  ' 
Changing  the  signs  a?=3. 


78  ELEMENTS  OF  ALGEBRA.  [sECT.  IV. 

Jfote  2. — To  verify  the  result  obtained  by  the  reduction 
of  an  equation,  substitute  the  value  obtained  for  the  unknown 
quantity  in  the  first  equation,  and  see  if  it  satisfies  the  con- 
ditions. Thus,  substituting  3  for  x  in  the  equation  above, 
we  have  -         -    4x3—5x3=9—12. 

Multiplying  factors  12—15  =  9—12. 

Reducing       -         -  — 3=  -^3. 

EXAMPLES. 

1.  Reduce  the  equation,  a7+_-|-_=  11.  Ans.  x^Q. 

2.  Reduce  the  equation  ^x-{- — 2——x-\--,     Ans.  a?=i. 

0  o 

3.  Reduce  the  equation IL_— 2=::1.  Ans.  x=1, 

4.  Reduce  the  equation  — -^-j- — —x-{-2.  Ans.  a?=4^. 

11  o 

5.  Reduce  the  equation  -+-+-  =  94.         Ans.  a;=120. 

3    4     5 


6.  Reduce  the  equation — + — =07 — 20 — - 
^  4'     10 


Ans.  a? =800. 

7.  Reduce  the  equation  ^^^+2&=.?^±i2. 

4  6 

Ans.x^^-:^^, 
9 

8.  Reduce  the  equation  8f +^±l=4+a:— 26i. 

5 

Ans.  07=39. 

n    -D    1        ^1  ^.       2a7— 5  ,  19— a?     10a7 — 7     5 

9.  Reduce  the  equation + = — ^-. 

^  18  3  9  2 

Ans.  07=7. 

10.  Reduce  the  equation  2o;— ?±i+15=l?^±?5 

o  0 

Ans.  07=12. 

11.  Reduce  the  equation  ^II?+?= 20—^11^. 

^  2       3  2 

Ans.  07=18. 


SECT.  IV.]       EQUATIONS    OP    THE    FIRST    DEGREE.  79 

12,  Reduce  the  equation  ?^±5— 5=1 


^7W.x=.^-"^ 


13.  Reduce  the  equation  _^^lt_=ca?4-4a. 


4 


2a^3c 
14.  Reduce  the  equation  Sb-^lax=3x-\-4ic — ex 

4c -8i 


^ns.  x= 


7a— 3+c 


15.  Reduce  the  equation  ^— ?H-?=20— -. 

1/6     9     3  2 


^»5.  a?=24|t. 


J7  .  X    a? 


16.  Reduce  the  equation  -+-— _4-a?=2a?— 43. 
4    5     6 


^715.  x=60. 


17.  Reduce  the  equation  3af+_ _=a7+a. 


6-hfr 

18.  Reduce  the  equation  -+-4--4-?— ?=1. 

2     3    4     5     6 

19.  Reduce  the  equation  3^-3_3x-4^^       27+4j? 

^  4    -        3'         '  9 

•^»*.  a?=9. 

20.  Reduce  the  equation  15^±i5-4=?^Ill?— 5. 

^  3j?+6  x—2 

Ans.  j:=2. 

21.  Reduce  the  equation  1^+^+1 2a  =10—?!::^. 

.^,.  .._115  +  156~180a 
23 

22.  Reduce  the  equation  31-^_|,^5^+8__7x— 8^^^ 

2  13  11 

Ans,  07=9. 


80  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

23.  Reduce  the  equation  5a?-4_3a:— 7^^i_8j:— 1 

^  6  10  ^         3 

24.  Reduce  the  equation  — I —  = +_. 

36  5a: — 4      4 

*dns.  a? =8. 

25.  Reduce  the  equation  !I^±?— 8=::?Zll?5f +4. 

^  3a:— 1  3a:— 1 

.>^7i5.    a:=:l. 

o«    T?    1        ^u  *•       2a?+l     402— 3a:     ^     471— 6a? 

26.  Reduce  the  equation — — =  9 — . 

^  29  12  2 

»dns.  x=12. 

27.  Reduce  the  equation  15^15+11^21^9^+15^ 

^  28  6a:+ 14  14 

j^ns.  07=7. 

28.  Reduce  the  equation  1?±^ :  3a:+6  : :  2 :  5. 

5 

^ns.  x=S. 

29.  Reduce  the  equation  3a?+25a :  9a?+4Z> : :  4  :  10. 

^ns.  .^250a-16&^ 
6 

30.  Reduce  the  equation  ??±25 . 7_3^ . .  jq  :  7. 

jlns.  a:=||. 

31.  Reduce  the  equation  21-3a?_4^+6^g_5^+l, 

^  3  9  4 

jlns.  a: =3. 

00    T?  J        *u  .•       6a:+8     5a:+3     27— 4a?     3a:+9 

32.  Reduce  the  equation 1— — ! — = — — -L_. 

^  11  2  3  2 

jSns.  x=z6. 

33.  Reduce  the  equation  a,^^^— 9g__5a:+2^g         2a?+5 

^  4  6  "         3 

12 


SECT.  IV.]      EQUATIONS    OF    THE    FIRST    DEGREE.  81 

o.    T,    1        *u  ♦•      7x— 8  ,  15x-i-8     o       31— X 

34.  Keduce  the  equation -f- ■ — =6X— — - — 

^  11  13  2 

Ans.  a: =9. 

35.  Reduce  the  equation  ^"^^  :  1 : : 2a:-f  19 :  3a:--19. 

^  6x— 43 

^ns.  x=8. 

36.  Reduce  the  equation  5j:+2!^±^^t^9-f  12^-1?. 

^715.  x  =  3. 

37.  Reduce  the  equation  ^^±^4- ^^i^=—-|-3U. 

^  25  9x-16      5        '^ 


38.  Reduce  the  equation 


^ns,  x=:4. 
4x—34_258— 5x^69— X 
17~  3  2 

^«^.  x=51. 
4x— 2    2x4-11     7— 8x 


39.  Reduce  the  equation  2x 

^  13  5  7 

*dns»  x=7. 

40.  Reduce  the  equation  16x4-5  :  ^^±i* : :  36x4- 10 : 1. 

^ns.  x=5. 

PROBLEMS   PRODUCING    EQUATIONS    OF    THE    FIRST   DEGREE,  IN- 
VOLVING   ONLY    ONE    UNKNOWN    QUANTITY. 

169.  Though  no  general  and  definite  rule  can  be  given 
for  the  translation  of  a  problem  into  algebraic  language,  yet 
the  following  precepts. may  be  found  useful  for  this  purpose. 

1.  Let  X  represent  the  unknown  quantity  whose  value  we  wish 
to  determine. 

2.  Indicate  by  the  aid  of  algebraic  signs  the  operations  that 
would  be  necessary  in  order  to  verify  the  answer  were  the  problem 
already  solved. 

3.  The  equation  or  proportion  thus  formed  may  be  reduced  by 
the  preceding  rules. 

PROBLEMS. 

1.  Two  men,  A  and  B,  trade  in  company  and  gain  $680, 
of  which  B  has  4  times  as  much  as  A.  What  is  the  share 
of  each  \ 

L 


82  ELEMENTS    OP   ALGEBRA.       1  ;  5.4      [SEGT.  IV. 

Let  x=  number  of  dollars  in  A's  share ; 

Then  4a7=  number  of  dollars  in  B's  share, 

And  we  shall  have  the  equation     a?+4a?=z680. 

Reducing  terms        -         -         -  5a:=680. 

Dividing  by  coefficient  of  a?      -  a?=136,  A's  share. 

And 4a?=::544,  B's  share. 

Verification  -         -    136+4x136  =  680. 

2.  What  number  is  that,  the  sum  of  whose  third  part  and 

fourth  part  is  7 1 

Let      0?=  the  number : 


Then   -=  one  third, 


And    -=z  one  fourth,  ^ 
4 


X       X 

And  we  shall  have  the  equation  _-4-_=7. 

3     4 

Clearing  of  fractions,  4a7+3a:=84  : 

Reducing  terms  -         7a7i=84: 

Dividing  by  7    -         -  a?=:12.  jSns, 

Verification    -  i^+l?=4+3=:7. 

3       4 

3.  Divide  $5000  between  A,  B,  C,  and  D  in  such  a  man- 
ner that  A  shall  have  $300  more  than  B,  and  B  $50  more 
than  C,  and  C  $|200  more  than  D.    W^hat  was  the  share  of 

each  ] 

Let  a?=  D's  share  ; 

,  Then  07+200=  C's  share; 

And   a?+250=  B's  share  ; 
And    a:+550=:  A's  share. 
And  we    shall  have   the   equation  a:+a7+200+a7+250+ 

a:+550 -1^5000:  . 
Transposing        -    a?+a?  +  a?+a?=5000— 200— 250-550; 
Reducing  -  4a?=4000; 

Dividing  by  4      -  a?=100  \  D's  share  : 

a?+ 200=  1200,  C's  share  : 
a?+ 250=  1250,  B's  share: 
,       07+550=1550,  A's  share. 


SECT.  IV.]       EQUATIONS    OF    TUE    FIRST    DEGREE.  83 

Verification,  1000-f- 1000+200+ 1000+250+ 1000+550= 

5000. 
Or,  reducing,  5000=5000. 

4.  It  is  required  to  divide  the  number  84  into  two  such 
parts  that  the  greater  shall  be  to  the  less  as  8  to  5. 

Let  x=  the  greater  part, 

And  84 — x=  the  less  part. 
And  we  have  the  proportion  x :  84 — x : :  8 :  5, 

Converting  the  proportion  )  ^x=612—Sx,  ' 

into  an  equation  ) 

Transposing         -         -         5a:+8x=672. 
Reducing  terms    -         -  13x=672. 

Dividing  by  13     -        -  0^=51/5,  greater  part. 

84— x=32i^,  less  part. 

5.  It  is  required  to  divide  $972  between  A  and  B  in  such 
a  manner  that  B  may  have  fths  as  much  as  A. 

Let       Xz=  A's  share, 

And   ^=  B's  share, 
5 

And  we  shall  have  the  equation  j?+ — =972. 

Clearing  of  fractions  -  5x+4x=4860. 

Reducing  terms  -         -  9a?=:4860. 

Dividing  by  9    -        -        -  a: =540,  A's  share. 

*^=432,  B's  share. 
5 

6.  A  man  puts  out  three  fifths  of  his  money  at  6  per  cent, 
and  the  remainder  at  7  per  cent.,  and  at  the  end  of  the 
year  receives  $4825  interest.  How  much  money  had 
hel 

Let        x=  the  amount : 

3a: 
Then  — =  the  amount  at  6  per  cent, 
0 

2x 
And     __=r  the  amount  at  7  per  cent., 
0 


84  ELEMENTS    OF   ALGEBRA.  [sECT.  IV. 

And,  multiplying  each  amount  by  its  rate,  we  shall  have 
the  equation 

5      100      5      100 

Multiplying  factors    —  +  1^=4825. 
^^    °  500^500 

Clearing  of  fractions  18a?4-  14a? =24- 12500 : 

Eeducing  terms     -         -     32a?=2412500 :   ^ 

Dividing  by  32       -         -         a7=75390|.  ^ns. 

7.  A  can  do  a  piece  of  work  in  8  days,;  B  can  do  the  same 
work  in  12  days;  in  what  time  will  they  do  it  if  both 
work  together  X 

A  will  do  ith  of  the  work  in  one  day  : 
B  will  do  yLth  of  the  work  in  one  day : 
Let   x=:  the  time  it  would  take    them  to  do  the  work 
which  is  represented  by  1  : 

Then,  in  x  days  A  will  do  -  of  the  work, 

And  in  x  days  B  will  do  —  of  the  work, 

^  12  '         • 

'  And  we  shall  have  the  equation  -+  —  =  1. 
^  8^12 

Reducing        -         -         -         -  a:=i:4f.  Ans, 

8.  A  gentleman  meeting  5  poor  persons,  distributed  $4,50 
among  them,  giving  to  the  second  twice,  to  the  third 

7     three  times,  to  the  fourth  four  times,  and  to  the  fifth 
five  times  as  much  as  to  the  first.     How  much  did  he 
.  give  to  each  1         Ans.  30,  60,  90,  120,  and  150  cents. 

9.  A  man  left  $11004  to  be  divided  among  his  widow,  two 
sons,  and  three  daughters,  in  such  a  manner  that  the 
widow  should  have  twice  as  much  as  both  the  sons,  and 
each  son  should  have  as  much  as  the  three  daughters. 
What  was  the  share  of  each  1 

Widow's  share,  $6288,  ) 

Each  son's  share,   $1572,         \  ^ns. 
Each  daughter's  share,  $524.  j 


SECT.   IV.]       EQUATIONS    OF    THE    FIRST    DEGREE.  85 

10.  What  number  is  that  which,  being  multiplied  by  8,  the 
product  increased  by  10  times  the  number,  and  that 
sum  divided  by  12,  the  quotient  shall  be  4 1  »^ns.  2*. 

11.  A  post  is  \  in  the  earth,  f  in  the  water,  and  13  feet 
out  of  the  water.     What  is  the  length  of  the  post  1 

•^716.  35. 

12.  After  paying  away  ]  and  4  of  my  money,  I  had  $85 
left  in  my  purse.     How  many  dollars  had  I  at  first  \ 

Jlns.  140. 

13.  Of  a  battalion  of  soldiers.(the  officers  being  included), 
I  are  on  duty,  ^^  sick,  f  of  the  remainder  are  absent, 
and  there  are  48  officers.  What  is  the  number  of  per- 
sons in  the  battalion  1  ,^7is,  800. 

14.  In  an  orchard  of  fruit-trees,  ^  of  them  bear  apples,  \ 
pears,  J  plums:  7  bear  peaches,  3  bear  cherries,  and  2 
quinces.     How  many  trees  are  there  1  ./^n*.  96. 

15.  A  farmer  being  asked  how  many  sheep  he  had,  an- 
swered, he  had  them  in  4  pastures:  in  the  first  he  had 
J  of  the  whole  number,  in  the  second  i,  in  the  third  ^, 
and  in  the  fourth  he  had  18  sheep.   How  many  had  he  1 

^ns.  72. 

16.  A  and  B  talking  of  their  ages,  A  says  to  B,  if  ^,  i,  and 
^  of  my  age  be  added  to  my  age,  and  2  years  more, 
the  sum  will  be  twice  my  age.     What  was  his  age  1 

Jlns.  84. 

17.  The  rent  of  an  estate  is  this  year  8  per  cent,  greater 
than  it  was  last.  This  year  it  is  $1890;  what  was  it 
last  yearl  .Ans.  $1750. 

18.  A  capitalist  receives  a  yearly  income  of  $2940  ;  |  of 
his  money  being  at  4  per  cent,  interest,  and  the  re- 
mainder at  5  per  cent.     How  much  has  he  at  interest! 

Ans.  $70,000. 

19.  A  cistern,  containing  60  gallons  of  water,  has  three 
unequal  cocks  for  discharging  it.  The  largest  will 
empty  it  in  1  hour,  the  second  in  2  hours,  and  the  third 
in  3  hours  In  what  time  will  they  empty  the  cistern 
if  they  all  run  at  once  1  Ans,  32,«j  minutes, 

8 


86  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

20.  A  farmer  wishes  to  mix  90  bushels  of  provender,  con- 
sisting of  rye,  barley,  and  oats,  so  that  the  mixture  may 
contain  |  as  much  barley  as  oats,  and  i  as  much  rye  as 
barley.  How  much  of  each  must  there  be  in  the  mix- 
ture % 

Ans.  50  bushels  of  oats^  30  of  barley^  and  10  of  rye. 

21.  A,  B,  and  C  trade  in  company.  A  puts  into  their 
stock  $3  as  often  as  B  puts  in  $7  and  C  $5.  They  gain 
$960.     What  is  each  man's  share  of  the  gain  '? 

Ans.  A's  $192,  B's  $448,  C's  $320. 

22.  A,  B,  and  C  trade  in  company.  A  puts  in  $700,  B 
$450,  and  C  $950.  They  gained  $420.  What  was  the 
share  of  each  % 

Ans.  A's  $140,  B's  $90,  and  Cs  $190. 

23.  At  a  certain  election,  the  whole  number  of  votes  was 
673.  The  candidate  chosen  had  a  majority  of  11.  How 
many  voted  for  each  1       Ans.  One  342,  the  other  331. 

24.  On  canvassing  the  votes  at  a  certain  election,  it  was 
found  that  there  was  no  choice  :  it  was  also  ascertained 
that  one  of  the  candidates  had  |  of  the  whole  number 
of  votes,  the  other  |  of  the  whole  number,  and  there 
were  45  scattering  votes.  What  was  the  whole  number 
of  votes  1  Ans.  200. 

25.  Three  men  built  780  rods  of  fence.  The  first  built 
9  rods  per  day,  the  second  7,  the  third  5 ;  the  second 
worked  three  times  as  many  days  as  the  first,  and  the 
third  twice  as  many  days  as  the  second.  How  many 
days  did  each  work  1 

26.  A  gentleman  bequeathed  $65,600  to  his  wife,  two 
sons,  and  three  daughters.  The  wife  was  to  have 
$2000  less  than  the  elder  son  and  $3000  more  than  the 
younger  son,  and  the  portion  of  each  of  the  daughters 
was  $3500  less  than  that  of  the  younger  son.  What 
was  the  share  of  the  wife  and  each  son  \ 

$16,350  elder  son's  share. 
$14,350  wife's  share.  ^^s. 

$11,350  younger  son's  share. 
$7,850  each  daughter's  share. 


SECT.  IV.]       EQUATIONS    OP   THE    FIRST    DEGREE.  87 

27.  A  man  meeting  some  beggars,  gave  each  of  them  4cf., 
and  had  I6d.  left  j  if  he  had  undertaken  to  give  them 
6d.  apiece,  he  would  have  wanted  12f/.  more  for  that  pur- 
pose. How  many  beggars  were  there,  and  how  many 
pence  had  he  1 

28.  A  boy  being  sent  to  market  to  buy  a  certain  quantity 
of  meat,  found  that  if  he  bought  beef,  which  was  4>d.  per 
pound,  he  would  lay  out  all  the  money  he  was  intrust- 
ed with ;  but  if  he  bought  mutton,  which  was  3^d.  per 
pound,  he  would  have  2  shillings  left.  How  much  meat 
was  he  sent  for  1 

29.  It  is  required  to  divide  85  into  two  such  parts  that 
I  of  the  one  added  to  ^  of  the  other  may  make  60. 

30.  When  the  price  of  a  bushel  of  barley  wanted  but  3d. 
to  be  to  the  price  of  a  bushel  of  oats  as  8  to  5,  four 
bushels  of  barley  and  90<i.  in  money  were  given  for 
nine  bushels  of  oats.  What  was  the  price  of  each  per 
bushel ( 

31.  A  market-woman  bought  a  certain  number  of  eggs  at 
the  rate  of  2  for  a  cent,  and  an  equal  number  at  3  for  a 
cent.  She  sold  the  whole  lot  at  the  rate  of  5  for  2  cents, 
and  lost  4-  cents  by  her  trade.  How  many  eggs  of  each 
sort  had  she  1  '  ^tis.  120. 

32.  The  hold  of  a  vessel  contained  442  gallons  of  water, 
which  was  emptied  out  by  two  buckets,  the  greater  of 
which,  holding  twice  as  much  as  the  other,  was  emp- 
tied twice  in  3  minutes,  but  the  less  3  times  in  2  min- 
utes, and  the  whole  time  of  emptying  was  12  minutes. 
How  much  would  each  bucket  contain  1 

Ans.  13  and  26. 

33.  The  expense  of  paving  a  square  court  at  50  cents  per 
square  yard,  is  the  same  as  that  of  surrounding  it  with 
an  iron  fence  at  $1,  75  per  foot.  How  many  square  feet 
does  it  contain  \  Ans.  15,876. 

34.  Out  of  a  certain  sum  a  man  paid  his  creditors  $96 ; 
half  of  the  remainder  he  lent  a  friend ;  he  then  spent 


88  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

one  fifth  of  what  remained,  and  after  all  these  deduc- 
tions had  one  tenth  of  his  money  left.  How  much  had 
he  at -first  1  ^ns.  $128. 

35.  There  are  two  numbers  whose  sum  is  a  sixth  part  of 
their  product,  and  the  greater  is  to  the  less  as  3  to  2. 
What  are  the  numbers  %  Arts,  10  and  15. 

36.  A  person  being  asked  the  hour,  answered  that  it  was 
between  5  and  6,  and  the  hour  and  minute  hands  were 
together.    What  was  the  time  \     Arts.  5A.  27;w.  16/y5' 

37.  Four  places  are  situated  in  the  order  of  the  four  let- 
ters, A,  B,  C,  and  D.  The  distance  from  A  to  D  is 
102  miles ;  the  distance  from  A  to  B  is  to  the  distance 
from  C  to  D  as  2  to  3;  and  -]  of  the  distance  from  A 
to  B,  added  to  \  of  the  distance  from  C  to  D,  is  three 
times  the  distance  from  B  to  C.  What  is  the  distance 
between  the  places  \  36  from  A  to  B  ) 

12  from  B  to  C  J.  ^j^^^ 
54  from  C  to  D  j 

38.  A  waterman  went  down  a  river  and  returned  again  in 
6  hours.  Now  with  the  stream  he  can  row  9  miles  an 
hour,  but  against  it  he  can  make  a  headway  of  only  3 
miles  an  hour.     How  far  did  he  gol    Ans.  13g  miles, 

39.  A  hare  is  50  leaps  before  a  hound,  and  takes  4  leaps 
to  the  hound's  3;  but  2  of  the  hound's  leaps  are  equal 
to  three  of  the  hare's.  How  many  leaps  must  the  hound 
take  to  catch  the  hare  1  Ans.  300. 

40.  There  is  a  certain  number  consisting  of  two  digits  or 
figures,  and  their  sum  is  6.  If  18  be  added  to  the  num- 
ber, the  sum  will  consist  of  the  same  digits  transposed. 
What  is  the  number  1  Ans.  24. 

J^Cote. — As  the  local  value  of  figures  increases  in  a  tenfold 
ratio  from  right  to  left,  if  x—  the  left-hand  digit,  and  6— 
a?=  the  right,  then  10a:+6 — a?  the  number. 

41.  A  man  and  his  wife  w^ould  consume  a  sack  of  meal 
in  15  days.     After  living  together  6  days,  the  woman 


SECT.  IV.]      EQUATIONS    OP   THE    FIRST   DEGREE.  89 

alone  consumed  the  remainder  in  30  days.     How  long 
would  the  sack  last  either  of  them  alone  1 

The  man,  21  ^   days.    ) 
The  woman,  50  days.  ) 

42.  In  the  composition  of  a  quantity  of  gunpowder,  the 
nitre  was  10  lbs.  more  than  |  of  the  whole  ;  the  sulphur 
4i  lbs.  less  than  ^  of  the  whole  ;  the  charcoal  2  lbs.  less 
than  4  of  the  nitre.  What  was  the  amount  of  gunpow- 
der 1  ^ns.  69  lbs. 

43.  Two  pieces  of  cloth,  of  the  same  price  per  yard,  but 
of  different  lengths,  were  bought,  the  first  for  JES,  the 
second  for  £Q^.  If  10  be  added  to  the  length  of  each, 
their  sums  will  be  as  5  to  6.  What  was  the  length  of 
each  piece  ]  •dns.  20  and  26. 

44.  A  and  B  began  trade  with  equal  sums  of  money.  The 
first  year  A  gained  JG40,  and  B  lost  JG40.  The  second 
year  A  lost  a  of  what  he  had  at  the  end  of  the  first, 
and  B  gained  JC40  less  than  twice  the  sum  which  A 
had  lost.  B  then  had  twice  as  much  money  as  A 
How  much  had  each  at  first  1  jJm.  £320. 

45.  On  an  approaching  war,  594  men  are  to  be  raised  from 
three  towns,  A,  B,  and  C,  in  proportion  to  their  popu- 
lation. The  population  of  A  is  to  that  of  B  as  3  to  5, 
and  the  population  of  B  is  to  that  of  C  as  8  to  7.  How 
many  men  must  each  town  furnish  1 

.  46.  A  shepherd,  in  time  of  war,  was  plundered  by  a  party 
of  soldiers,  who  took  ]-  of  his  flock  and  {  of  a  sheep  j 
another  party  took  i  of  what  he  had  left  and  i  of  a 
sheep  ;  then  a  third  party  took  ^  of  what  remained  and 
i  of  a  sheep,  after  which  he  had  25  sheep  left.  How 
many  had  he  at  first  1  *^7is.  103. 

47.  A  merchant  adds  yearly  to  his  capital  one  third  of  it, 
but  takes  from  it,  at  the  end  of  each  year,  $500  for  his 
expenses.  At  the  end  of  the  third  year,  after  deduct- 
ing the  last  $500,  he  finds  his  original  capital  is  doub- 
led 1  What  was  that  capital  1  ^na.  $5550. 
M 


90  ELEMENTS    OF   ALGEBRA.  [sECT.  IV. 

48.  A  labourer  was  hired  for  48  days :  for  each  day  he 
wrought  he  was  to  receive  24s.,  but  for  each  day 
he  was  idle  he  was  to  forfeit  125.  At  the  end  of  the 
time  he  received  504*.    How  many  days  did  he  work  1 

^  Ans.  30. 

49.  A  cistern  which  holds  820  gallons,  is  filled  in  20  min- 
utes by  3  pipes,  one  of  which  conveys  10  gallons  more, 
and  the  other  5  gallons  less  than  the  third,  per  minute. 
How  much  flows  through  each  pipe  per  minute  I 

Arts.  22,  7,  and  12  gallons, 

50.  A  sets  out  from  a  certain  place,  and  travels  at  the 
rate  of  7  miles  in  5  hours  j  and  8  hours  afterward  B 
sets  out  from  the  same  place,  and  travels  the  same  road 
at  the  rate  of  5  miles  in  3  hours.  How  long  and  how 
far  must  B  travel  before  he  overtakes  A  1 

Ans.  42  hours,  and  70  miles, 

EUUATIONS  OF  THE  FIRST  DEGREE  INVOLVING  MORE 
THAN  ONE  UNKNOWN  aUANTITY. 

170.  Most  of  the  problems  which  we  have  already  con- 
sidered, involve  more  than  one  ufaknown  quantity  ;  but  we 
have  been  able  to  solve  them  by  employing  but  one  letter  or 
symbol,  as  we  have  found  it  easy,  by  means  of  this  letter 
and  the  conditions  of  the  problem,  to  find  expressions  for  the 
other  unknown  quantities.  In  many  cases,  however,  the 
solution  is  simplified  by  representing  more  than  one  of  the 
unknown  quantities  by  a  letter,  and  in  complicated  problems 
it  is  frequently  necessary  to  do  this. 

171.  When,  by  the  conditions  of  the  problem,  two  or  more 
unknown  quantities  are  to  be  determined,  it  is  necessary 
that  there  should  be  as  many  independent  equations  as  there 
are  unknown  quantities.  When  this  is  not  the  case,  the  prob- 
lem will  be  indeterminate.  Equations  are  independent  when 
they  express  different  conditions,  and  dependent  when  they 
express  the  same  conditions  under  different  forms. 

172.  Elimination  is  a  method  of  deriving  from  the  given 
equations  a  new  equation,  from  which  all  the  unknown  quan- 


SECT.  IV.]      EQUATIONS    OF   THE    FIRST    DEGREE.  91 

titles  except  one  shall  be  excluded.  The  unknown  quanti- 
ties thus  excluded  are  said  to  be  eliminated ;  and  the  re- 
sulting equation  may  be  solved  by  the  principles  already  dis- 
cussed and  applied.  Having  found  the  value  of  one  of  the 
unknown  quantities,  the  others  may  be  readily  found  by 
substitution. 

173.  There  are  three  principal  methods  of  elimination,  viz., 
1.  By  Compariion;  2.  By  Substitution  ;  3.  By  Addition  or 
Subtraction. 

OF    ELIMINATION    WHEN   THERE    ARE    TWO    EQUATIONS   INVOLVING 
TWO    UNKNOWN    QUANTITIES. 

First  Method. — By  Comparison, 

174.  This  method  of  elimination  rests  upon  the  axiom, 
that  if  each  of  two  things  is  equal  to  a  third,  they  are  equal  to 
each  other. 

Let  us  take  the  two  equations  -        x-\-y=16y 

2a?+3y=36. 
Finding  the  value  of  x  in  the  1st  equation,    a7:=16— y, 

Finding  the  value  of  x  in  the  2d  equation,     x= — ^^^. 

i 

Since  each  of  the  two  quantities,  16 — y,  and  ^,  is 

equal  to  x,  they  must  be  equal  to  each  other  (Ax.). 

Hence  we  have  the  equation,      -         16 — y— '    "~"  ^ 

Reducing y=4'.    • 

Substituting  for  y  its  value  in  the  1st  equation,  x-|-4=  16, 
Reducing a?=16 — 4<=12. 

175.  Hence,  for  the  elimination  of  one  of  two  unknown 
quantities,  by  comparison,  we  have  the  following  general 

RULE. 

1.  Find  the  value  of  one  of  the  unknown  quantities  in  each  of 
the  equations. 

2.  Form  a  new  equation  by  placing  these  two  values  equal  to 
each  other. 


92  ELEMENTS    OF    ALGEBRA.  [sECT.   IV. 

J^ote. — It  will  generally  be  found  convenient  to  eliminate 
that  unknown  quantity,  which  is  least  involved  with  known 
quantities. 

EXAMPLES. 

1.  Reduce  the  equations  ^x-\-y—^^  and  4y-f-a;=16. 

4a7-f-y=:34', 

37+4^=16, 

Finding  the  value  of  x  in  the  1st  equation,  a?= y, 

Finding  the  value  of  x  in  the  ^d  equation,    x=  16 — 4y, 

Forming  a  new  equation  -  __JZ^=:16 — 4y, 

Reducing           -----        y=2, 
And a?r=8. 

2.  Reduce  the  equations  2a;-j-3y=16^,  and  3a; — 2y=ll. 

i^ns,  a;=:5,  andyz=z2. 

3.  Reduce  the  equations  _+2l— 7,  and  -+^=8. 

^  2     a      '         3     2 

^ns.  x  —  6j  and  y=12, 
a — h 


4>.  Reduce  the  equations  _+2y=a,  and  - — ^y=b 


4 


Jlns.  a;=a+Z>,  and  y-. 

5.  Reduce  the  equations  --|-^=:8,  and  - — ^^=1. 

iL         O  O  At 

Ans.  a:=12,  and  y=6, 

6.  Reduce  the  equation  --\-±=:9^  ^nd  the  proportion  x: 

2^  -  4  :  3.  ^ns.  a:=  12,  and  y=9. 

7.  Reduce  the  equations  — ±—£.=8 — -,  and  -llZ — =rll 

^  6  3  2 

-\-y.  Ans.  a?=6,  and  y=S. 

Second  Method. — By  Substitution. 
176.  This  method  of  elimination  rests  upon  the  principle, 
that  if  any  equivalent  expression  be  substituted,  in  an  equa- 
tion, for  an  unknow^n  quantity,  it  will  satisfy  the  conditions 
of  that  equation. 


SECT.  IV.]       EQUATIONS    OF   THE    FIRST    DEGREE.  93 

Let  US  resume  the  two  equations  before  used, 

2x-f3y=36. 
Finding  a  value  of  j:in  the  1st  equation,  x=16 — y: 

Substituting  this  value  for  x  in  the  2d  equation, 

2(16— y)4-3y=36; 

Reducing  -        - V—^i 

And x=12. 

177.  Hence,  for  the  elimination  of  one  or  two  unknown 
quantities  by  substitution,  we  obtain  the  following  general 

RULE. 

1.  Find  the  value  of  one  of  the  unknown  quantities  in  one 
of  the  equations. 

2.  In  the  other  equation^  substitute  this  value  for  the  unknovm 
quantity  itself  and  then  reduce  as  before. 

EXAMPLES. 

1.  Reduce  the  equations  a;+y=13,  and  x— y=3. 

Jlns,  37=8,  andy=z6, 

2.  Reduce  the  equations x — 7=3y— 21,  and  x-{-l=2y+  H, 

Ans.  a:=:49,  andy=^\, 

3.  Reduce  the  equations  7a:=8y,  and  a?— y+^O. 

Ans.  x=  160,  and  y=  140. 

4.  Reduce  the  equations  ic4-10=2y,  and  y-j-10=3x. 

jlns.  x  —  6y  and  y=:S. 

5.  Reduce  the  equations  ^+8y=194,  and  l-\-Sx=zl3h 

o  8 

jJns.  07=16,  andy  —  2^. 

6.  Reduce  the  equations  4a7+^=26,  and  -4-^=6. 

2  2    5 

Ans.  x=4,  andy=^20. 

7.  Reduce  the  proportions  a? :  y  : ;  3  : 1,  and  - :  5y— 4:  :3:9. 

Ans.  a: =24,  and  y=S. 

8.  Reduce  the  equations  ?±?-|-6y=21,  and  y4-^+5a:=23. 

4  3 

Ana,  x=4,j  and  y=3^. 


94  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

Third  Method. — By  Addition  or  Subtraction, 
178.  This  method  of  elimination  rests  upon  the  prmciple, 
that  if  equals  be  added  to  or  subtracted  from  equals,  the  re- 
sults will  be  equal. 

As  the  members  of  an  equation  are  equal  quantities,  it 
will  follow  that  if  one  equation  be  added  to  or  subtracted 
from  another,  the  results  will  be  equal. 

Let  us  resume  the  two  equations  before  used, 

2a? -I- 33^^  36. 
Multiplying  the  1st  equation  by  2  -       2a7H-2y=32  5 

Subtracting  the  3d  from  the  2d  equation  y=  4; 

Substituting  and  reducing      -         -         -  3?=:  12. 

Again,  let  us  take  the  two  equations,        3r+5y=:28, 

2a?— 5y=  2. 


Adding  the  two  equations  -  -  5a?  =30; 
Dividing  by  5  -         -         -         -a?         =6j 

Substituting  and  reducing  -  '  y  =2. 
179.  Hence,  for  the  elimination  of  one  or  two  unknown 

quantities,  by  Addition  or  Subtraction,  we  have  the  following 

general 

RULE. 

1.  Multiply  or  divide  the  equations  in  such  a  manner  that  the 
term  containing  one  of  the  unknown  quantities  shall  be  the  same 
in  both  equations. 

2.  If  the  signs  of  these  terms  are  alike,  subtract  one  equation 
from  the  other  ;  if  unlike,  add  the  two  equations  together. 

EXABIPLES. 

1.  Reduce  the  equations  2x-{-y—16,  and  3a: — 3y=6. 

Ans.  x=6,  and  y=4f. 

2.  Reduce  the  equations  a?+y=48,  and  x—y=z32. 

Ans.  a?=40,  andy=S. 

3.  Reduce  the  equations  5a?— 3yi=9,  and  2x-\-by=16. 

Ans.  07=3,  and  y—2. 


SECT.  IV.]       EQUATIONS    OF    TUB   FIRST    DEGREE.  95 

4.  Reduce  the  equations  30x-\-^0yz^210,  and  50xH-30y= 
340.  Jins.  x=b,  and  y=3. 

5.  Reduce  the  equations  3a?-}-7y=79,  and  2y=9-f  ?. 

^ns.  x=z  10,  and  y=7. 

6.  Reduce  the  equations  ?fZ:??4-2=7,  and  ?+?-:=6. 

y         o  5 

^n*.  a;  1=24,  and  y=  10, 

7.  Reduce  the  equations  ?fZ^4-14=18,  and  r^+f-f  16 

2  3 

=  19.  ^715.  x=i5,  and  y=2. 

8.  Reduce  the  equations  ^+^"^^+^^8,  and  llZ^l-y^ 
11.  ^ns,  x  =  6y  and  y=S, 

9.  Reduce  the  equations  '^_}0-^ ^9^:10  ^^^  2y+4 

^  5  3  4'  3 

8  4 

10.  Reduce  the  equations  _+?l=:6,  and  -4-^=5|. 

,^ns,  x=  12,  and  y  =  16. 

MISCELLANEOUS    EQUATIONS   OF    THE    FIRST  DEGREE  INVOLVING 
TWO   UNKNOWN    QUANTITIES. 

1.  Reduce  the  equations  8x  +  5y=68,  and  12x+7y=100. 

,^ns.  a: =6,  and  y=4. 

2.  Reduce  the  equations  8x4-^=20,  and  20j:-|-3y=70. 

^dns.  x=\ljand  y=15. 

3.  Reduce  the  equations  £±2^— 2y=2,  and  '^~~^y-\-y= 

23  ' 

— -.  ^ns,  x=  1 1,  and  y=l. 

Ot       Q 

4.  Reduce  the  equations  — Z_-|-y=7,  and  5x — 13y=y 

•  ,^ns,  x=S,  andy  =  ^. 


96  ELEMENTS    OF    ALGEBRA.  [seCT.   IV. 

5.  Reduce  the  equations  ?^2^=?^y±i,  and  8-^!=^ 

3  5  5 

=  6.  Jins.  a:=13,  and  y=zS, 

6.  Reduce  the  equations  ^+?^=— ,  and  ^-\-'?l=^%\. 

^  5       4      20'  4       5      ''^ 

jlns.  a?=i,  and  yz=:^» 

7.  Reduce  the  equations  ?+ 7^=99,  and  ^  +  7a?zzi51. 

»^ns^  a? =7,  a?itZ  yz=  14. 

8.  Reduce  the  equations  ?— 12=^+8,  and  ^±y+^— 8= 

^y~^+27.  Jins.  a?=60,  a«(/  v=40. 

4  ^ 

9.  Reduce  the  equations  ^—12=^+13,  and  ^±2(_}_^_f_i6 

=^~y+27.  ^«5.  a? =60,  awc^  v=20. 

4 

10.  Reduce  the  equations    "^  — — I^i=3v — 5,  and    ^ 

^  5  4^'  2 

4,  J, 3 

,    4- — ^ — =18 — 5a;.  Ans.  x=^^  and  y—% 

6 

11.  Reducethe  equations  4a?+l^Z:?=2y+5+2!^±ii,and 

12.  Reduce  the  equations  1 — ^  .    +^—y — 16|,  and  ^^ — 2 
=-.  ^»5.  07=  10,  and  y=20. 

13.  Reduce  the  equations  ??II?l(=a:— 2|,  and  a?— ?(ZL^=:0. 

./^ws.  a?=l,  andy—Z. 

14.  Reduce  the  equations  ^ — ^-+5  =  6,  and  3'-f-4= — +6. 

4     7  5  14 

./?;j5.  a: =28,  and  y =20. 


15.  Reduce  the  equations  y — 3=-  +  5,  and    ~^^=y — 32 
•  ^?^5.  a?=2,  awc?y=9. 


19 


SECT.  IV.]       EQUATIONS    OF    THE    FIRST    DEGREE.  97 

x-l-3  3x ^v 

16.  Reduce  the  equations  2y — _I__=7-f- — — ^,  and  4x — 

4  5 

^Z:y=2^—^^^.  Ans.  x=5,  and y=5. 

17.  Reduce  the  equations  X— ?yz:?±f  =1-1- i5f±iy,  and 

*  11  33     ' 

3x-f2y_y— 5_llx+152_3y-H 
6  4  12  2    ' 

Arts,  a; =8,  andy=9. 

18.  Reduce  the  equations  ?^?^=18i— i^?^,and 

15  *  7        ' 

10y+^^~~^^  =  5^+lQx.  Ans.  x=:10,  and  y  =16. 

5 

.  Reduce  the  equations  g_3j?-h5y     i7^5y^^+7  ^^^ 

17  ^3 

22--6y_5x-7^x4:l  _8y+5^     ^^  ^^^ 

3  11  6  18  '        ^ 

20.  Reduce  the  equations  — ~~'"  4-  ^~  =4+ — ^^^^ — »and 
^  6  3  2 

2j+y     9a:— 7     3y  +  9     4r-|-5y     ^  a       j       a 

— 21^— ^=J^-— ^.  ^/w.  j;=9,  an(/y=4. 

PROBLEMS  REQUIRING  TWO  UNKNOWN  QUANTITIES,  AND  PRODUCING 
TWO  EQUATIONS  OF  THE  FIRST  DEGREE. 

181. — 1.  A  fruiterer  sold  to  one  person  6  lemons  and  3 
oranges  for  42  cents,  and  to  another  3  lemons  and  8 
oranges  for  60  cents.     What  was  the  price  of  each  1 
Let  x=  the  price  of  a  lemon. 
And  y=  the  price  of  an  orange, 
Then  we  shall  have  the  two  equations, 

6x-|-3y=42, 
3x-h8y=60. 
Transposing  in  the  Ist  equation      -        -    6a:=42 — 3y; 

Dividing  by  6 ^^42— 3y.. 

6 
Transposing  in  the  2d  equation       -        -    3a7=60— 8y; 

Dividing  by  3 ^^60-8y. 

9  N 


98  ELEMENTS    OF   ALGEBRA.  [sECT.  IV. 

Forming  a  new  equation  from  the  two  values  of  x, 
42— 3y_60— Sy 

—6 3-"' 

Reducing  -         -         y=6,  the  price  of  an  orange, 

And  -         -         -         x—4fj  the  price  of  a  lemon. 

2.  What  fraction  is  that,  to  the  numerator  of  which,  if  1 
be  added,  its  value  will  be  i,  but  if  1  be  added  to  the  de- 
nominator, its  value  will  he  1% 

Let     x=  the  numerator. 
And  y=  the  denominator, 

Then  -=  the  fraction, 

y 

And  we  shall  have  the  two  equations, 

x-^1 


=h 


—1. 


y+1    ' 

Clearing  the  1st  of  fractions       -         3x-\-3=y; 
Clearing  the  2d  of  fractions        -         -      4>x—y-{-li 

Dividing  the  4th  equation  by  4  -         -        rrr=?Li_ ; 

4 

Substituting  the  value  of  a? )         /y+l\  ,  «_ 

in  the  third  equation      -  y^  \    4   /        ~^ ^ 

Reducing y=15, 

And  ----._  07=4 : 

Hence -=tt5  the  frac- 

y 

tion  required. 
3.  A  boy  bought  2  apples  and  3  oranges  for  13  cents;  he 
afterward  bought,   at   the   same   rate,  3  apples  and  5 
oranges  for  21  cents.     What  was  the  price  of  eachl 
Let    x=  the  price  of  an  apple, 
And   y=  the  price  of  an  orange. 
Then  we  shall  have  the  two  equations, 

2:c+3y=13, 
3x+53/=:21. 


SECT.  IV.]       EQUATIONS    OF  THE    FIRST    DEGREE.  99 

Multiplying  the  Ist  equation  by  3,  6x-f-9y=39; 
Multiplying  the  2d  equation  by  2,  6a:+10y=4.2j 
Subtracting  the  3d  from  the  4th        -        y=3,  the  price 

of  an  orange, 
And  - a?=2,  the  price 

of  an  apple. 

4.  What  fraction  is  that,  to  the  numerator  of  which  if  4 
be  added,  the  value  is  i,  but  if  7  be  added  to  its  de- 
nominator, the  value  is  }  1  Ans.  y\. 

5.  A  and  B  have  certain  sums  of  money :  says  A  to  B, 
"Give  me  $15  of  your  money,  and  I  shall  have  five  times 
as  much  as  you  have  left."  Says  B  to  A,  "  Give  me  $5 
of  your  money,  and  I  shall  have  exactly  as  much  as  you 
have  left."     How  many  dollars  had  each  1 

Ans.  Jl  had  $35,  and  B  $25. 

6.  There  are  two  numbers,  such  that  3  times  the  greater 
added  to  ^  the  less  is  equal  to  36 ;  and  if  2  times  the 
greater  be  subtracted  from  6  times  the  less  and  the  re- 
mainder divided  by  8,  the  quotient  will  be  4.  What 
are  the  numbers  1  Ans,  ^  and  l\, 

7.  A  person  was  desirous  of  relieving  a  certain  number 
of  beggars  by  giving  them  25.  6f/.  each,  but  found  that 
he  had  not  money  enough  in  his  pocket  by  3*.:  he  then 
gave  them  2^.  each,  and  had  4s.  to  spare.  How  many- 
shillings  had  he,  and  how  many  beggars  did  he  relieve  1 

Ans.  32*.  and  14  beggars, 

8.  A  labourer  working  for  a  gentleman  for  12  days,  and 
having  had  with  him  the  first  7  days  his  wife  and  son, 
received  745. :  he  wrought  afterward  8  other  days,  du- 
ring 5  of  which  he  had  with  him  his  wife  and  son,  and 
he  received  505.  Required  the  gain  of  the  labourer  per 
day,  and  also  that  of  his  wife  and  son. 

Ans.  Husband  55.,  and  the  wife  and  son  Is, 

9.  A  purse  holds  19  crowns  and  6  guineas.  Now  4  crowns 
and  5  guineas  fill  yy  of  it.  How  many  will  it  hold  of 
each  %  Ana,  21  crovms  and  63  guineas. 


100  ELEMENTS  OF  ALGEBRA.  [sECT.   IV. 

10.  A  farmer,  with  28  bushels  of  barley  at  25.  4c/.  per 
bushel,  would  mix  rye  at  3s.  per  bushel^,  and  wheat  at  4>s. 
per  bushel,  so  that  the  whole  mixture  may  consist  of 
100  bushels,  and  be  worth  3^.  4c?.  per  bushel.  How 
many  bushels  of  rye,  and  how  many  of  wheat,  must  he 
mix  with  the  barley  %      Ans.  20  of  rye  and  52  of  wheat. 

11.  A  and  B  speculate  with  different  sums  :  A  gains  $150, 
B  loses  $50,  and  now  A's  stock  is  to  B's  as  3  to  2. 
And  if  A  had  lost  $50,  and  B  gained  $100,  then  A's 
stock  would  have  been  to  B's  as  5  to  9.  What  was  the 
stock  of  each  1  Ans.  A's.  $300,  and  B's  $350. 

12.  A  rectangular  bowling-green  having  been  measured, 
it  was  observed  that,  if  it  were  5  feet  broader  and  4  feet 
longer,  it  would  contain  116  feet  more  j  but  if  it  were 
4  feet  broader  and  5  feet  longer,  it  would  contain  113 
feet  more.     Required  the  length  and  breath. 

Ans.  Length  12,  and  breadth  ^  feet. 

13.  There  is  a  number  consisting  of  two  figures,  the  sec- 
ond of  which  is  greater  than  the  first  j  and  if  the  num- 
ber be  divided  by  the  sum  of  its  figures,  the  quotient  is 
4 ;  but  if  the  figures  be  inverted,  and  the  number  which 
results  be  divided  by  a  number  greater  by  2  than  the 
difference  of  the  figures,  the  quotient  becomes  14.  What 
is  the  number  1  Ans.  48. 

14.  A  person  owes  a  certain  sum  to  two  creditors.  At  one 
time  he  pays  them  $53,  giving  to  one  y^  of  the  sum 
due  to  him,  and  to  the  other  $3  more  than  ^  of  his  debt 
to  him.  At  a  second  time  he  pays  them  $42,  giving 
to  the  first  ^  of  what  remains  due  to  him,  and  to  the 
other  A  of  what  is  due  to  him.     What  were  the  debts  \ 

Ans.  $121  and^m. 

15.  Some  smugglers  discovered  a  cave  which  would  ex- 
actly hold  the  cargo  of  their  boat,  viz.,  13  bales  of  cat- 
ton  and  33  casks  of  wine.  While  they  were  unloading, 
a  custom-house  cutter  coming  in  sight,  they  sailed  away 


SECT.  IV.]       EQUATIONS  OF    THE    FIRST    DEGREE.  101 

with  9  casks  and  5  bales,  leaving  the  cave  |  full.    How 
many  bales  or  casks  would  it  hold  1 

Ana.  24  halts  or  72  casks. 
16.  A   and  B  can  perform  a  piece  of  work  in   16  days. 
They  work  together  4  days  j  then  A  being  called  off,  B 
is  left  to  finish  it,  which  he  does  in  36  days  more.     In 
what  time  would  each  do  it  separately  1 

Ans.  A  in  24,  and  B  in  48  days. 
V7,  Two  loaded  wagons  were  weighed,  and  their  weights 
were  found  to  be  in  the  ratio  of  4  to  5.  Parts  of  their 
loads,  which  were  in  the  proportion  of  6  to  7,  being 
taken  out,  their  weights  were  then  found  to  be  in  the 
.  ratio  of  2  to  3  j  and  the  sum  of  their  weights  was  then 
10  tons.     What  were  their  weights  at  first  1 

Ans.  16  and  20  tons. 

18.  There  is  a  cistern,  into  which  water  is  admitted  by 
three  cocks,  two  of  which  are  of  exactly  the  same  di- 
mensions. When  they  are  all  open,  j^  of  the  cistern  is 
filled  in  4  hours  ;  and  if  one  of  the  equal  cocks  be  stop- 
ped, ^  of  the  cistern  is  filled  in  lO^.  hours.  In  how  many 
hours  would  each  cock  fill  the  cistern  1 

Ans.  Each  of  the  equal  ones  in  32  hours,  and  the  other 
in  24. 

19.  A  has  a  capital  of  $30,000,  which  he  puts  out  to  in- 
terest at  a  certain  rate  per  cent.,  and  he  owes  $20,000, 
on  which  he  pays  a  certain  rate  per  cent,  interest*  The 
interest  which  he  receives  exceeds  that  which  he  pays 
by  $800.  B  has  a  capital  of  $35,000,  which  he  puts  out 
to  interest  at  the  same  rate  per  cent,  that  A  paysj  he 
also  owes  $24,000,  on  which  he  pays  interest  at  the 
same  rate  that  A  receives.  The  interest  which  he  re- 
ceives exceeds  that  which  he  pays  by  $310.  What  are 
the  two  rates  of  interest  1  Ans.  6  and  5  per  cent. 

20.  A  has  a  certain  capital,  which  he  puts  out  to  interest 
at  a  certain  rate  per  cent.  B  has  a  capital  of  $10,000 
more  than  A,  which  he  puts  out  to  interest  at  one  per 
cent,  more,  and  receives  $800  more  interest  than  A. 


0  ^  '  v-^ 


102  ELEMENTS  OF  ALGEBRA.  [sECT.  IV. 

C  has  a  capital  of  $15,000  more  than  A,  which  he  puts 
out  at  2  per  cent,  more,  and  receives  $  1500  more  interest 
than  A.  What  is  the  capital  of  each,  and  the  three  rates 
of  interest  1  Jlns.  A's  capital,  $30,000  ;  B's,  $40,000  ; 
C'5,  $45,000  ;  and  the  rates  of  interest  4,  5,  and  &  per  cent, 

Ot  ELIMINATION  WHERE  THERE  ARE  THREE  OR  MORE  EQUATIONS 
INVOLVING  AS  MANY  UNKNOWN  QUANTITIES. 

182.  In  the  problems  hitherto  given,  each  has  contained 
no  more  than  two  unknown  quantities,  and  two  independent 
equations  have  been  sufficient  to  express  the  conditions  of 
the  question.  Other  problems,  however,  may  involve  three 
or  more  unknown  quantities  j  and  if  they  are  determinate, 
their  conditions  will  give  rise  to  as  many  independent  equa- 
tions as  there  are  unknown  quantities. 

183.  The  principles  already  discussed,  and  the  rules  al- 
ready given  for  the  elimination  of  one  of  two  unknown 
quantities,  may  also  be  applied  where  the  number  exceeds 
two. 

Thus,  if  there  be  three  independent  equations  involving 
three  unknown  quantities, 

I.  From  the  three  equations  involving  three  unknown  quanti- 
ties, deduce  two  equations  involving  only  two  unknown  quantities. 

II.  Then  from  these  two  deduce  one,  involving  only  one  un- 
known quantity. 

III.  Reduce  this  equation,  or  find  the  numerical  or  literal 
value  of  the  unknown  quantity  involved  in  it :  then  substitute 
this  value  for  the  unknown  quantity  itself,  in  an  equation  which 
involves  only  that  and  another  unknown  quantity  whose  value 
may  thus  be  found.  The  value  of  the  remaining  unknown  quan- 
tity may  be  found  in  a  similar  manner. 

184.  If  there  be  four  independent  equations  involving  four 
unknown  quantities, 

I.  From  the  four  equations  deduce  three,  involving  only  three 
u?iknown  quantities. 

II.  Reduce  these  three  equations  as  before. 

185.  If  there  be  n  independent  equations,  involving  n  un- 


SECT.  IV.]   EQUATIONS  OF  THE  FIRST  DEGREE. 


103 


known  quantities,  they  maybe  reduced  in  a  similar  manner. 
For  from  the  n  equations  involving  n  unknown  quantities, 
we  may  deduce  n — 1  equations  involving  n— 1  unknown 
quantities ;  and  from  these  n — 2  equations  involving  n — 2 
unknown  quantities,  and  so  on  until  only  one  equation  re- 
mains, involving  only  one  unknown  quantity.  The  value  of 
this  being  found,  the  values  of  all  the  rest  may  be  determin- 
ed by  substitution,  as  before. 

A  calculation  may  often  be  very  much  abridged  by  the 
exercise  of  judgment  in  stating  the  question,  in  selecting 
the  equations  from  which  others  are  to  be  deduced,  in  the 
manner  of  performing  the  reduction,  in  simplifying  fractional 
expressions,  in  avoiding  radical  quantities,  &c. 


EXAMPLES 


1.  Reduce  the  equations 


a?-f-  y+  Z-. 
x-\-2y-\-32z 

U  U  -■- 

2       3^  4 


29' 

62 

.    Jlns. 

10 

, 

, 

From  the  1st  equation 
From  the  2d  equation 
From  the  third  equation 

Making  the  1st  and  2d  values  ) 
of  X  equal  -        -  V         ^ 

Making  the  1st  and  3d  values 
of  X  equal 

From  the  7th  equation 

From  the  8th  equation 


x=S, 

2=12. 


x=29—y—z;  (4) 
a:=62— 2y— 3z;(5) 
x=20-?^— ^;  (6) 


J29-y- 


Making  the  two  values  of  y 

equal         -         -         - 


33- 


Keducing 


Substituting  for  z  its  value 

the  9th  equation 
Substituting  for  y  and  z  their 

values  in  the  4th  equation 


1 


3 

2 

-z=62— 2y- 

-3z.  (7) 

^z=20-^- 

-I-    (8) 

y=33-2z; 

(9) 
(10) 

2.=27_|. 

(11) 

z=n; 

(12) 

y= 33—24  = 

=9.(13) 

x=29— 9- 

12=8. 
(14) 

104 


ELEMENTS    OF   ALGEBRA. 


[sect.  IV. 


[2x^^—3z=22)  (x=3, 

2.  Reduce  the  equations  {  4<x~2y+5z=  18  \    ^^^^  \  y=l, 

[Qcc+ly—  z=Q3]  [z=^. 


Multiplying  the  1st  by  6 
Multiplying  the  2d  by  3 
Multiplying  the  3d  by  2 


12a?-h24y-18z:=132. 
12a:— 62/+15z=54. 
12a?+14y— 22r=126. 


Subtracting  the  5th  from  the  4th  30y — 33z=:78. 
Subtracting  the  5th  from  the  6th  20y— 17:^=72. 
Multiplying  the  7th  by  |  -  -  20^—222=52. 
Subtracting  the  9th  from  the  8th^  -         52=20. 

Dividing  by  5       -         -         -         -         -  2=4, 

And y—'^i 

And j?=3. 


(4) 
(5) 
(6) 
O) 
(8) 
(9) 
(10) 

(11) 
(12) 
(13) 


3.  Reduce  the 
equations 


Ans. 


fl2a?4-     y+  7m=26 
182+    3y+ 12^=69 
1007  +  202+17^=69 
18^+102+  7y=66^ 
From  the  1st  equation,  ^=26 — lu — 12ir. 
Substituting  for  y  ") 

its  value  in  the  [    182+78— 21w—36a?+12M=69 
2d   -         -  j 

Substituting  for  y ") 

its  value  in  the  \  18ir+  IO2+ 182— 49 w—84a:= 66.       (7) 
4th  -    .     -  J 


2=li, 
w=2. 

(5) 
(6) 


Transposing    and  ) 

uniting  in   the  \ 

6th   -         -  J 

Transposing   and  "j 

uniting   in  the  J- 

7th  -         -  J 

Multiplying  the  9th  by  2,    2O2— 98w— 132^?=— 232. 
Subtractingthe  10th  from  the  3d,  115M+142a?=301. 
Multiplying  the  8th  by  5,    902— 45w— 1800?=  -45. 
Mdtiplying  the  9th  by  9,  902— 441t^ 
Subtracting  the  13th  ) 

from  the  12th  \ 


I82— 9w— 36a:=-9.     (8) 


IO2— 49w— 66a7=— 116.     (9) 

(10) 

(11) 

(12) 

594a;=— 1044.(13) 

396M+414a;=999.       (14 


^  SECT.  IV.]       EQUATIONS    OF    THE    FIRST    DEGREE. 

999— 396m 


From  the  14lh  equation,  x= 


414. 


From  the  11th  equation,  i=2?il"il^. 
^  '  14-2 

Making  these  two  values  (  999— 396t<_ 301— 115m 


of  X  equal 


4U 


142 


105 
(15) 
(16) 
(17) 


Reducing 
And 
And 
And 


tt=2. 


4.  Reduce  the  equations 


^    y    " 


^ns. 


:23,V. 


121 


^y  *  'r 

Adding  the  three  equations,  -+Z-|-z=.j.-f.|-f-_'_z=l:!ll.  (4) 

X       y       Z  OQy) 

1+1+1^121.  (5) 
X     y     z     720    ^  '^ 

31      ^  ^    720    ... 

: or   2:  = ,    (O) 

z     720'  31'  ^  ^ 

Subtracting  the  2d  from  the  5th,    -=  11,  or  v=— ,  (7) 

'    y     720'        ^      41  '  ^  '^ 

Subtracting  the  3d  from  the  5th,    -=—,  or  x=— .  (8) 

X     720'  49     ^  ^ 


Dividing  by  2       - 

Subtracting  the  1st  from  the  5th, 


5.Eeducethe(^2y-3.=  4,l 

equations       |  yi"-'  >  r 

t   a?-f  y4-  2=12.  j 

6.  Reducethe|^^+^y+  ^=*^' ] 
equations         i^"  y+^^=  ^' 

r    x+  y-J-   z-l-   w=14,' 


7.  Reduce  the 
equations 


3x+2y-|2r-f  w=lb, 
2. 


^     2y       z_3^. 
3       4      5 


^ns. 


Ans, 


Am, 


rx=5, 

rx=2o, 

^y  =  12, 
iz=:32 
'  x=2, 

y=3, 

2  =  4, 

w=i5. 


106 


ELEMENTS    OF    ALGEBRA. 


[sect.  IV. 


8.  Reduce  the 
equations 


a?+yr-.52,  ^  (  x=:20,. 

2/+  2^=82,  3/=:32, 

z-\-w=iQS^  >  Jljf,s,\  z—bO, 

u-{-x='62.j  [  u—12. 

9.  Reduce  the  equations  a?i/=28,  a?2r=2(>,  and  yz=35. 

Ans.  a:=4i,  y— 7,  and  z=z5. 


10.  Reduce  the 
equations 


2^3^4 


124, 


^  3    4     5  '  ^ 


Ans, 


L4     5     6' 


76. 


:48, 


y=120, 


:^=:240. 


11.  Reduce  the  equations  a?y=100,  ^0=40,  and  a?2=160. 

Ans.  07=20,  y  =  5,  z—'^, 

12.  Reduce  the  equations  ir+100=2/H-2r,  y4-100=2a?+ 
22;,  and  2;+100=i3a?+3y. 

Ans.  a?:=9Jyj  2/=45/y,  aw  J  2r=63yV 


13.  Reduce  the 
equations 


107+ 32/=:  23,) 

4 
2/+32r  =  31, 

^a?+2/+i2r+2w;=35.  J 


Ans. 


2     3     4' 


:62, 


14.  Reduce  the  J  ^_|_y_j_^_47 
equations  o     ^     o 

4     5     6 

2a?  +  2^—22:= 

4y — a?-j-3zi 

3w+w: 

3a?— yH-3w — w: 


Ans. 


15.  Reduce  the 
equations 


:40, 
:35, 
:13, 
:15, 

=49. 


Am.  < 


16.  Reduce  the 
equations 


[00+  y+  z=:  53,] 
\  a?+22/-h32:=105,  \ 
i^a?  +  32/+4z=134.J 


a?=6, 

y=% 

z=S, 
a?=24, 

y-60, 

^=120. 

a? =20, 
y  =  10, 

z=5, 

u^l. 
fa?=24, 


Ans.  J  II  — 


^2=23. 


SECT.  IV.]       EQUATIONS    OF    THE    FIRST    DEGREE.  107 

PROBLEMS    PRODUCING    THREE    OR    MORE    EQUATIONS,  AND 
REQUIRING    AS    MANY    UNKNOWN    QUANTITIES. 

1.  Three  persons  divided  a  sum  of  money  between  them 
in  such  a  mapner  that  the  shares  of  A  and  B  together 
amounted  to  $900 ;  the  shares  of  A  and  C  together 
amounted  to  $800  j  and  the  shares  of  B  and  C  to  $700. 
What  was  the  share  of  each  1 

Let  a7=A's  share, 
y=B's  share, 
z=C^s  share: 
Then  -        x+y=900y 
And    -        j?+2=&00, 
And    -        y+2=700. 
Reducing,         x=500,  A's  share, 
y=400,  B's  share, 
2=300,  C's  share. 

2.  A  man  with  his  wife  and  son,  talking  of  their  ages, 
said  that  his  age  added  to  that  of  his  son  was  16  year& 
more  than  that  of  his  wife ;  the  wife  said  that  her  age 
added  to  that  of  her  son  made  8  years  more  than  that 
of  her  husband  ;  and  that  all  their  ages  together  amount- 
ed to  88  years.     What  was  the  age  of  each  1 

Ans.  Husband  40,  wife  36,  and  son  12  years, 

3.  Three  teachers,  A,  B,  and  C,  speaking  of  their  respect- 
ive schools,  says  A  to  B,  "If  you  will  give  me  20  of 
your  scholars,  my  number  will  then  be  to  the  sum  of 
C's  and  what  you  will  have  left,  as  4  to  5."  Says  B  to 
A  and  C,  "  If  each  of  you  will  give  me  10  scholars,  my 
number  will  be  to  what  you  will  then  have  as  5  to  4." 
Says  C  to  A  and  B,  "  If  you  will  give  me  10  each,  I 
shall  have  twice  as  many  as  both  of  you."  What  was 
the  number  of  scholars  each  had  1 

Ans,  A  20,  B  30,  and  C  40  scholars, 

4.  A  cistern  is  furnished  with  three  pipes,  A,  B,  C.  By 
the  pipes  A  and  B  it  can  be  filled  in  12  minutes,  by  the 
pipes  B  and  C  in  20  minutes,  and  by  A  and  C  in  15 


108  ELEMENTS    OF    ALGEBRA.  [sECT.  IV. 

minutes.  In  what  time  will  each  fill  the  cistern  alone, 
and  in  what  time  will  it  be  filled  if  all  three  run  to- 
gether 1  Ans.  A  20,  B  30,  C  60  minutes^  and  the  three 
together  in  10  minutes. 

5.  It  is  required  to  divide  the  number  72  into  4  such  parts, 
Uiat  if  the  first  part  be  increased  by  5,  the  second  di- 
minished by  5,  the  third  part  multiplied  by  5,  the  fourth 
part  divided  by  5,  the  sum,  difference,  product,  and 
quotient  shall  all  be  equal.  Ans.  5,  15,  2,  and  50. 

6.  Find  three  numbers,  such  that  \  of  the  first,  \  of  the  sec- 
ond, and  \  of  the  third  shall  be  equal  to  62  ^  i  of  the  first, 
\  of  the  second,  and  ]  of  the  third  equal  to  47  j  and  \ 
of  the  first,  \  of  the  second,  and  \  of  the  third  equal  to 
38.  Ans.  24,  60,  and  120. 

7.  If  A  and  B  together  can  perform  a  piece  of  work  in 

8  days,  A  and  C  together  in  9  days,  and  B  and  C  in  10 
days,  how  many  days  will  it  take  each  person  alone  to 
perform  the  same  work  ? 

Ans.  A  in  14f|,  B  in  17-^, -and  C  in  23/^  days. 
.  8.  A,  B,  and  C  sit  down  to  play,  each  one  with  a  certain 
number  of  shillings :  A  loses  to  B  and  C  as  many  shil- 
lings as  each  of  them  has.  Next  B  loses  to  A  and  C  as 
many  as  each  of  them  now  has  :  lastly,  C  loses  to  A  and 
B  as  many  as  each  of  them  now  has.  At  the  close  of 
the  game,  each  of  them  has  16  shillings.  How  much 
did  each  one  gain  or  lose  1 

Ans.  A  lost  10s.,  B  gained  25.,  and  C  Ss. 

9.  There  are  two  such  fractions,  that  if  3  be  added  to  the 
numerator  of  the  first,  its  value  is  double  that  of  the 
second ;  but  if  3  be  added  to  the  denominator,  their 
values  are  ecfiial.     Now  the  sum  of  the  two  fractions  is 

9  times  as  great  as  their  difference  ;  and  if  the  numera- 
tor of  their  product  be  increased  by  10,  its  value  will 
be  equal  to  that  of  the  first  fraction.  What  are  the 
fractions'?  Ans.  j\  and  ^. 

10.  Three  brothers  purchased  an  estate  for  $15,000:  the 


SECT.  IV.]       EQUATIONS    OF    THE    FIRST    DEGREE.  109 

first  wanted,  in  order  to  complete  his  part  of  the  pay- 
ment, ^  of  the  property  of  the  second  j  the  second 
would  have  paid  his  share  with  the  help  of  ^  of  what 
the  first  owned ;  and  the  third  required,  to  make  the 
same  payment,  in  addition  to  what  he  had,  |  part  of 
what  the  first  possessed.  What  was  the  amount  of  each 
one's  property  1 

Jlns.  $3000,  $4000,  and  $4250  respectively. 

11.  A  merchant  has  3  ingots,  each  composed  of  gold,  sil- 
ver, and  copper,  in  the  following  proportions,  viz.,  in  the 
first  there  are  7  ounces  of  gold,  8  ounces  of  silver,  and 
1  ounce  of  copper  to  the  pound;  in  the  second,  there 
are  5  ounces  of  gold,  7  ounces  of  silver,  and  4  ounces 
of  copper;  and  in  the  third,  2  ounces  of  gold,  9  ounces 
of  silver,  and  5  ounc^  of  copper  to  the  pound.  What 
parts  must  be  taken  from  each  in  order  to  compose  a 
fourth  ingot,  in  which  there  shall  be  4}  f  ounces  of  gold, 
7||  ounces  of  silver,  and  3j^  ounces  of  copper  to  the 
pound  1 

Ans,  4  ounces  ofgoldj  9  ounces  of  silver,  and  3  ounces 
of  copper. 

12.  At  an  election  for  two  members  of  Congress,  three 
men  offer  themselves  as  candidates:  the  number  of  vo- 
ters for  the  two  successful  ones  are  in  the  ratio  of  9  to 
8;  and  if  the  first  had  had  7  more,  his  majoritj'^  over  the 
second  would  have  been  to  the  majority  of  the  second 
over  the  third  as  12  to  7.  Now  if  the  first  and  third 
had  formed  a  coalition,  and  had  one  more  voter,  they 
would  each  have  succeeded  by  a  majority  of  7.  How 
many  voted  for  each  1 

Jlns.  369,  328,  and  300  respectively. 
10 


110  .ELEMENTS    OF    ALGEBRA.  [sECT.  V. 


SECTION  V. 

Generalization  of  Algebraic  Problems. — Demonstration  of  Gen- 
eral  Propositions  or  Theorems. — Properties  of  J^umhers. — 
Reduction  of  Formulas  relating  to  Simple  Interest^  Com- 
pound  Interest^  and  Fellowship, — Discussion  of  Equations 
of  the  First  Degree. — Theory  of  Negative  Quantities. — Ex- 
planation of  Symbols. — Infinity. — Infinitesimal. — Indetermi' 
nation. — Inequations. 

GENERALIZATION  OF  ALGEBRAIC  PROBLEMS. 

188.  The  soiution  of  many  questions  does  not  depend 
upon  the  particular  numbers  given  in  those  questions,  but 
will  be  the  same  for  any  other  numbers.  By  generalizing 
such  questions,  we  are  able  to  deduce  a  general  method  or 
rule  for  the  solution  of  all  questions  whose  conditions  are 
similar,  or  which  differ  from  the  proposed  only  in  particular 
numbers  which  are  given. 

The  following  instances  of  generalization  will  serve  to  in- 
troduce the  learner  into  this  important  branch  of  Algebra. 
First  General  Problem. 

189.  The  sum  of  two  numbers  is  a,  their  difference  b;  it 
IS  required  to  find  the  two  numbers. 

Let  x=  the  greater,  and  y=  the  less: 
Then,  by  the  conditions  -         -        x-\-y—a. 

And        -         -         -         s        -        -        X — y:=b ; 

Adding  the  two  equations       -        -       2a?       =a-\-b, 

2     2' 
Subtracting  the  2d  from  the  1st  equation,  2y=a—b; 

_a b 

^~2     2* 
Hence,  since  a  and  b  may  represent  any  numbers  what- 
ever, the  sum  and  difference  of  two  quantities  being  given, 


SECT,  v.]  GENERALIZATION.  Ill 

1.  To  find  the  greater,  ^dd  the  half  difference  to  the  half  sum, 

2.  To  find  the  less,  Subtract  the  half  difference  from  the  half 
sum. 

EX^AMPLES. 

1.  The  sum  of  two  numbers  is  24,  the  difference  6 :  what 
are  the  two  numbers  1 

Let  x=  the  greater,  and  y=  the  less : 

Then  x=^+|=?iH-^=  124-3=15,  the  greater, 
JL     JL      JL      JL  • 

And     y=-— -=——-=  12—3=9,  the  less. 
^2222  ' 

2.  The  sum  of  two  numbers  is  56,  their  difiTerence  12: 
what  are  the  numbers  \ 

3.  It  is  required  to  divide  $860  between  two  men,  so  that 
the  first  may  have  $250  more  than  the  second. 

4.  Two  merchants  invest  in  trade  $10,000;  the  sum  in- 
vested by  the  first  exceeds  that  invested  by  the  second 
by  $1225;  what  was  the  sum  invested  by  eachl 

Second  General  Problem. 
190,  The  sum  of  three  numbers  is  a  ;  the  excess  of  the 
mean  above  the  least,  b  ;  and  the  excess  of  the  greatest  above 
the  mean,  c.     Required  the  three  numbers. 

Let  x=  the  least,  y=  the  mean,  and  z=  the  greatest: 
Then,  by  the  conditions      -        x-\-y-\-z—a; 

y—x=b; 
z-^=c  ; 

Reducing  these  three  equations,  x=       ^  ? 

o 


a-\-b — c 
3 

a4-*-f-2c 


J'=-3-' 


2  = 

3 
Hence,  since  a,  i,  and  c  may  represent  any  values  what- 
ever, having  given  the  sum  of  three  numbers,  the  excess  of 
the  mean  above  the  least,  and  the  excess  of  the  greater 
above  the  mean  : 


112  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

1.  To  find  the  least,  Fro^  their  sum  subtract  the  sum  of  twice 
the  mean  above  the  least,  and  the  excess  of  the  greater  above  the 
mean,  and  divide  the  remainder  by  3. 

2.  To  find  the  mean,  To  their  sum  add  the  excess  of  the  mean 
above  the  least,  and  from  the  result  take  the  excess  of  the  greatest 
above  the  mean,  and  divide  the  remainder  by  3. 

3.  To  find  the  greatest,  Jldd  together  the  sum  of  the  three 
numbers,  the  excess  of  the  mean  above  the  least,  and  twice  the  ex- 
cess of  the  greatest  above  the  mean,  and  divide  the  sum  by  3. 

EXAMPLES. 

1.  The  sum  of  three  numbers  is  440 ;  the  excess  of  the 
mean  above  the  least  is  40 ;  the  excess  of  the  greatest 
above  the  mean  is  60  :  vi'hat  are  the  numbers  1 
Let  X,  y,  an^  z.  represent  the  numbers. 

Then  x=g=m=^^°-(-^^"+^°)=100,  the  least 

number ; 

a4j_e^440+40-60^^^Q^^  ^^^^^^^^ 
^33' 

_«4.ft+2c^440+40+2x60^200,  the  greatest 


3  3 

number. 

2.  It  is  required  to  divide  a  prize  of  $973  among  3  men, 
so  that  the  second  shall  have  $69  more  than  the  first, 
and  the  third  $43  more  than  the.  second. 

3.  The  sum  of  three  numbers  is  15,730  ;  the  second  ex- 
ceeds the  third  by  2320,  and  the  first  exceeds  the  sec- 
ond by  3575 :  what  are  the  three  numbers  1 

Third  Genera^  Problem. 
191.  The  sum  of  4  numbers  is  a;  the  second  exceeds  the 
first  by  b ;  the  third  exceeds  the  second  by  c ;  the  fourth 
exceeds  the  third  by  d.     Required  the  four  numbers. 

EXAMPLE. 

Find  each  of  the  above  numbers,  on  the  supposition  that 
a=:3753,  Z>=159,  c     275,  and  t/^389. 


SECT,  v.]  GENERALIZATION.  ]  13 

Fourth  General  Problem. 

192.  The  sum  of  2  numbers  is  a,  and  if  3  times  the  first 
be  divided  by  2  times  the  second,  the  quotient  will  be  b,' 
Bequired  the  numbers. 

EXAMPLE. 

If  a=420  and  6=&,  what  are  the  numbers  1 
Fifth  General  Problem, 

193.  The  sum  of  two  numbers  is  a,  and  if  the  first  be  di- 
vided by  5  and  the  second  by  2,  the  sum  of  the  quotient 
will  be  b.    Required  the  numbers. 

EXAMPLE. 

If  a=120  and  J=42,  what  are  the  numbers! 
Sixth  General  Problem, 

194.  Three  men  share  a  certain  sum  in  the  following  man- 
ner, viz.:  the  sum  of  A's  and  B*s  shares  is  a;  that  of  A's 
and  C*8,  b ;  that  of  B's  and  C's,  c.  What  is  the  sum  di- 
vided, and  the  share  of  each  % 

EXAMPLES. 

1.  If  a=:$123,  5= $110,  and  c=$83,  what  will  be  the  sum, 
and  the  share  of  each  1 

Seventh  General  Problem.  ' 

195.  A  person  engaged  a  workman  to  labour  n  days';  for 
each  day  that  he  laboured  he  was  to  receive  a  cents,  and  for 
each  day  he  was  idle  he  was  to  pay  b  cents :  at  the  time  of 
settlement  he  received  c  cents.  How  many  days  did  he  la- 
bour, and  how  many  was  he  idle  \ 

Let  x=  number  of  days  he  laboured, 
y=  number  he  was  idle ; 


114  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

Then,  by  the  conditions, 


ax — hy=^c 


or  x=z  — L_, 
a-i-b' 

1        an — c 
and  y: 


a+b 

JVote. — If  the  labourer  had  paid  c  cents  instead  of  receiv- 
ing it,  the  general  equations  would  become, 

bn — c 


a!-\-y=n 
by—ax=zc 


or  0?: , 

a-\-b' 

J         an-{-c 
and  y= ! — 


EXAMPLES. 

1.  Jf  w=48,  a=24,  b—-12,  and  c=r504,  how  many  days 
did  he  work,  and  how  many  was  he  idle  1 

2.  A  labourer  was  hired  for  75  days  :  for  each  day  that  he 
wrought  he  was  to  receive  $3,  but  for  each  day  that  he 
was  idle  he  was  to  forfeit  $7.  At  the  time  of  settle- 
ment he  received  $125  :  how  many  days  did  he  labour, 
and  how  many  was  he  idle  1 

3.  A  man  agreed  to  carry  20  earthen  vessels  to  a  certain 
place  on  this  condition,  viz.,  that  for  every  one  deliver- 
ed safe  he  should  receive  11  cents,  and  for  every  one 
he  broke  he  should  forfeit  13  cents :  he  received  124 
cents.     How  many  did  he  break  1 

4.  A  fisherman,  to  encourage  his  son,  promises  him  9 
cents  for  each  throw  of  the  net  in  which  he  should  take 
any  fish  j  but  the  son  is  to  forfeit  5  cents  for  each  un- 
successful throw.  After  37  throws  the  son  receives 
from  the  father  235  cents.  What  was  the  number  of 
successful  and  unsuccessful  throws  of  the  net  1 

DEMONSTRATION    OF    GENERAL    PROPOSITIONS    OR  THEOREMS. 

196.  It  was  remarked  in  the  introductory  section  of  this 
work,  that  algebraic  symbols  might  be  applied  to  the  dem- 
onstration of  general  truths  or  principles.  We  will  now 
exhibit  a  few  of  these  applications. 


SECT,  v.]  THEOREMS.  115 

First  Theorem. 

197.  The  greater  of  any  two  numbers  is  equal  to  half 
their  sum  added  to  half  their  difference,  and  the  less  is 
equal  to  half  their  sum  minus  half  their  difference. 

Let  a  and  h  represent  any  two  numbers,  of  which  a  is  the 
greater  and  h  the  less  j  let  their  sum  be  represented  by 
5,  and  their  difl^erence  by  d: 

Then a+b=s. 

And a—b=d; 

Adding  the  equations     -        -        -      2a      =s-{-d;\ 

Dividing a      =i  +  ^-j 

Subtracting  the  2d  from  the  1st  equation,  2b=s — d;  ^ 

Dividing  -       ^ *=|— -.  I 

Second  Theorem. 

198.  The  product  of  the  sum  and  difference  of  two  num- 
bers is  equal  to  the  difl^erence  of  their  squares. 

Let  a,  b,  8,  and  d  sustain  the  same  relations  as  in  the  p(re- 
ceding  theorem : 

Then s=a-\-b, 

And      -        -        -        .        -        d=a — b. 


Multiplying  the  two  equations,  d.s  z=(a-\-b)(a — b)=a* — 6*. 
CoBOL.  1. — Dividing  the  above  equation  by  d,  we  have 
a»— A« 

Hence,  if  the  difllerence  of  the  squares  of  two  numbers  be 
divided  by  the  diflJerence  of  the  numbers,  the  quotient  will 
be  their  sum. 

CoROL.  2. — Dividing  the  same  equation  by  Sy  we  shall  have 


d-. 


a 


-I^ 


8 

Hence,  if  the  difference  of  the  squares  of  two  numbers 
be  divided  by  the  sum  of  the  numbers,  the  quotient  will  be 
their  difference. 


116  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 


Third  Theorem, 
199.  Four  times  the  product  of  any  two  numbers  is  equal 
to  the  squares  of  their  sum,  diminished  by  the  square  of 
their  difference. 

Let  a,  &,  5,  and  d  sustain  the  same  relations  as  in  the  pre- 
ceding theorem : 
Then        ------       a-\-b=s^ 

a—b=d. 


Adding  the  two  equations         -         -  2a—s-\-d; 

Subtracting  the  2d  from  the  1st         -  2b=zs — d; 

Multiplying  the  3d  and  4th        -         -         4}ah=zs^ — dK 

Fourth  Theorem.     - 

200.  The  sum  of  the  squares  of  any  two  numbers  is  equal 
to  the  square  of  their  difference  plus  twice  their  product. 

Let  a,  &,  and  d  sustain  the  same  relations  as  before,  and 
let  p  represent  the  product  of  the  two  numbers-: 

Then -         a — b=:d, 

And  ------  ab—p; 

Squaring  the  members  of  the  1st )         o  a  i  /2     ^. 

equation  "         "         "  ) 

Multiplying  the  2d  equation  by  2,  2a&       =2p/ 

Adding   the   two    equations   to-)  2_j_r2_-72io 

gether     -        -        -         -         j 

Fifth  Theorem, 

201.  The  square  of  a  polynomial  expressing  the  sum  of 
two  numbers,  is  equal  to  the  square  of  the  first  term  + 
twice  the  product  of  the  two  terms  +  the  square  of  the 
last  term. 

Let  s  represent  the  sum,  and  a-{-b  the  polynomial: 

Then s  =a -\-b ; 

Squaring  the  equation         -         -         s^—a^+'^ab-\-W. 

Sixth  Theorem, 

202.  The  square  of  a  polynomial  expressing  the  differ- 


SECT,  v.]  THEOREMS.  117 

ence  between  two  numbers,  is  equal  to  the  square  of  the 
first  term  —  twice  the  product  of  the  two  terms  -f-  the 
square  of  the  last  term. 

Let  d  represent  the  difference,  and  a—b  the  polyno- 
mial : 

Then d=a—b; 

Squaring  the  equation         -         -         (P=a" — 2a&+J* 

Seventh  Theorem. 
203.  The  difference  of  any  two  equal  powers  of  different 
numbers,  is  always  divisible  by  the  difference  of  the  num- 
bers. 

Let  a  and  b  represent  finy  iwo  numbers,  a  being  greater 
than  b : 

Then ^II^=a-\-b, 

a  — b 

And ^II^=(^+ab-\-l^, 

a  — b 

And ^LZI^^a^^a'b+ab'+k 

a — b 

This  process  may  evidently  be  continued  indefinitely; 
hence  we  have 

^!lll^=a'^'+a™-2x  J  +  a'"-='x&«-f- a^b'^-{-ab^-^-\-b'^K 

a  — b 

CoROL.  If  J=l  in  the  above  formula,  the  formula  will  be- 
come 

a"*— 1 


=a'^'+a'"-'4-a'"~^4-  ....  -f  o'+a'+a-fl. 


Eighth  Theorem, 

204.  The  difference  of  two  equal  powers  of  different  num- 
bers, is  divisible  by  the  sum  of  the  numbers,  when  tlie  expo- 
nent of  the  power  is  an  even  number. 

Let  a  and  b  sustain  the  same  relations  as  before : 

Then    -        .        .        ^Ill=a-b, 
a-\-b 


118  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

And      -         -         -         '^^II^=^a^—d?h+ah^—h\ 
a  -\-b 

And      -         -         -         ^^^=a'—a''b+aW—a'}y'-\-ab*—b\ 
a  -j-b 

Hence  we  also  conclude  (letting  m  represent  any  even 
number), 

- — ^=a^-'-j-d^^xb+ar-^Xb^-^  . . .  -j-a'b^'-^-^-ab^-^+b^K 
a  -^b 

CoROL.  If  ^=1,  the  above  formula  will  become 

«"» 1 

t==a^''-'a^-''+a"'-^—  ....  -^a^—a'-\-a—l. 

a  +1 

JSPinth  Theorem. 

205.  The  sum  of  any  two  equal  powers  of  different  num- 
bers, is  divisible  by  the  sum  of  the  numbers,  when  the  expo- 
nent of  the  power  is  an  odd  number. 

Let  a  and  b  sustain  the  same  relations  as  before : 

Then    -         -         '^±^=a'—ab-]-b% 
a  -\-b 

And      -         -         '!L±^=:a'—a'b+a'b^—ab^+b\ 
a  -{-b 

And      -         -         '^-±^=a'—a'b-\-a'b'—a'b'+a'b^—a¥+b\ 
a  -j-b 

Hence  we  also  conclude  (letting  m  represent  any  odd 
number), 

^-t!!l=a^~'—a^^xh-{-a""^Xb^—  ....  — a='6— "^-a^Z^'^-^— 
a  -\-b 

ab^''-\-b^-\ 
CoROL.  If  &=!,  the  above  formula  will  become 

a  +1 

Tenth  Theorem, 

206.  If  a  given  number  be  divided  into  two  parts,  and 
those  parts  multiplied  together,  the  product  will  be  the 
greatest  possible  when  the  parts  are  equal. 

Let  n=:  the  given  number,  a=.  the  greater  part,  b=.  the 


8BCT.  v.]  PROPERTIES    OF    NUMBERS.  119 

less  part,  d=  the  difference  between  the  parts,  and^= 
the  product  of  the  two  parts: 

Then -        -        ab=p. 

And  (Art.  189)  -        -  -        -        a  =^  +^, 

n     d 
2     2 


And  (Art.  189)  -        •        -'      -        -         J=s— -• 


Multiplying  the  last  two  equations  together,       ab=— — — . 

XT  n^    d* 

Hence -         P=— — -• 

^     4     4f 

Now  It  is  evident  that  p  will  increase  as  d  diminishes; 
hence  it  will  be  the  greatest  possible  when 

d=Oy  or  p=i    . 

DEMONSTRATIONS  RELATING  TO  CERTAIN  PROPERTIES  OF  NUMBERS. 

207.  We  will  now  apply  the  principles  of  Algebra  hereto- 
fore discussed  to  the  demonstration  of  some  singular  prop- 
erties of  numbers. 

Let  it  first  be  premised  that  the  local  value  of  the  digits 
increases  in  a  tenfold  ratio  from  right  to  left,  and  that  any 
number  is  equal  to  the  number  of  units  expressed  by  the 
digit  in  the  unit's  place,  -f  the  number  of  units  expressed 
by  the  digit  in  the  ten's  place,  -|-  the  number  of  units  ex- 
pressed by  the  digit  in  the  hundred's  place,  -f  j  &c. 

Thus,  3756  =  6  +  504-700+3000,  and  12,899=9+90+800 
+2000+10,000. 

>  First  Proposition, 

208.  If  from  any  number  the  sum  of  its  digits  be  subtract- 
ed, the  remainder  will  be  divisible  by  9. 

Let  a,  6,  c,  (/,  &c.,  represent  the  digits  of  any  number,  a 
being  the  digit  in  the  unit's  place,  h  the  digit  in  the 
ten's  place,  &c. ;  also  let  N=:  the  number,  n=  the  num- 
ber of  the  digits,  S=  the  sum  of  the  digits,  and  r=10: 


120  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

Then    'N=a-\-br  -j-cr"  +c?r'  +...+a?r"-', 

And       S=za-{-b  +c  +d  +...+a?. 

N— S=      br—b    -{-cr'—c    +d7^—d    +  . . . +a?r"-'-— a?, 
Or  N— S==      ^>(r— l)  +  c(r^— l)H-c/(r'— 1)+-  •  •  +x(r"-'— 1), 
by  subtraction. 

But  (Th.  7,  Art.  203)  r— 1,  r^-l,  r»— 1+  . .  +r"-^— 1,  are 
divisible  by  r — 1,  which  is  equal  to  9  j  hence,  N — S  is  also 
divisible  by  9. 

Example.  327,856  — (3+2+7+8+5  +  6)  =  327,825,  and 
.327,825^  9  ==36,425. 

Second  Proposition, 

209.  If  the  sum  of  the  digits  of  any  number  be  divisible 
by  9,  the  number  itself  is  divisible  by  9. 

Let  N=  any  number,  and  S=  the  sum  of  its  digits: 

Then,  since  S  is  supposed  to  be  divisible  by  9,  let  S=9m 

Since  N— S  is  divisible  by  9,  let  'N—S=9p  : 

Then  -         -         N— S=N— 9w  =  9j9; 

Transposing      -         -  N=z9p-{-9m ; 

Resolving  into  factors,  N=9(7?-}-m),  which  is  divisi 

ble  by  9 ;  consequently,  N  also  is  divisible  by  9. 

Example.  5 1,489 ^9=^5721,  and  (5+1+4+8+9)^9=27 
H-9  =  3.      ^ 

Third  Proposition. 

210.  If  the  sum  of  the  digits  of  any  number  be  divisible 
by  3,  the  number  itself  is  divisible  by  3. 

Let  N  represent  any  number,  and  S  the  sum  of  its  digits, 

as  before  ;  and  let  S  =  3m,  and  N — S=3p : 
Then    -         -     N— 37^  =  3;?  ; 

Transposing,  N=3p  +  3?7i,  which  is  evidently  divisible 
by  3. 

Example.  785,142-^3=261,714,  and  (7+8+5+1+4+2) 
-f-3=27H-3=9. 

Fourth  Proposition. 

211.  If  from  any  number  the  sum  of  the  digits  standing 
in  the  odd  places  be  subtracted,  and  to  it  the  sum  of  the 


SECT,  v.]  PROPERTIES    OP    NUMBERS.  121 

digits  Standing  in  the  even  places  be  added,  the  result  will 

be  divisible  by  11. 

Let  the  number  be      a+br         -f-cr*  -{-drf  Sec, : 

Add      -         -  — a+ft  — c  -fc/,  &c. 

The  result  is  -  br-^b    +cr'—c    ■i-dr^-{-d,    &c., 

Or         -         -         -  6(r+l)-|-c(r'— l)+c/(r'+l),  &c. 

But  (Ths.  8  and  9)  r'{- 1,  r»— 1,  r'-f- 1,  &:c.,  are  divisible  by 

r-f- 1 ;  hence,  b{r-\-  l)+c(r»— l)+t/(r'4- 1),  &c.,  is  divisible  by 

r+1,  or  11. 

Example.  (57,937-(7-t-94-5)  +  (7+3))-Ml =(57,937— 21 
H-10)-H  11=57,926-7-11  =  5266. 

Fifth  Proposition. 

212.  If  the  sum  of  the  digits  standing  in  the  even  places 
in  any  number  be  equal  to  the  sum  of  the  digits  standing  in 
the  odd  places,  the  number  is  divisible  by  11. 

Let  N=  the  number,  S=  the  sum  of  the  even  digits,  and 

$z=  the  sum  of  the  odd  digits  : 
Then,  by  Prop.  4,  N+S— s  is  divisible  by  11.     But  S— « 
=0j  therefore  N  is  divisible  by  11. 
Example.  (137,456+13- 13)^11  =  137,4.56-M1=12,496. 
Sixth  Proposition. 

213.  Every  prime  number  which  will  exactly  divide  the 
product  of  two  factors,  one  of  which  is  also  a  prime  num- 
ber, will  divide  the  other. 

Let  AxB  represent  the  product  of  two  numbers,  which 
is  divisible  by  P ;  A  being  greater  than  P,  and  prime 
with  it,  or  not  divisible  by  it. 

Then  let  us  endeavour  to  find  the  greatest  common  divi- 
sor of  A  and  P,  representing  the  successive  quotients 
^y  Qj  Q  >  &c*>  ai*d  the  successive  remainders  by  R,  R', 

11  Q 


122  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

P)A(Q 
QP 

"IR)  P  (Q' 
Q'R 

R')  R  (Q" 
Q"R 

R",  &c. 
Making  each  dividend  equal  to  the  product  of  its  divisor 
and  quotient,  we  have 

1.  -        -        -        A=PQ+R; 

2.  -        -        -         P=RQ'+R'; 

3.  -        -        -        R=:R'Q"+R",  &c. 
MuUiplying  the  first  equation  (1.)  by  B, 

AB=PQB  +  RBj 
Dividing  by  P       .        .        -        .        ^=  BQ+^. 

By  hypothesis,  _ -  produces  a  whole  number  ;  and  since 

B  and  Q  are  whole  nuhibers,  the  product  BQ  is  a  whole 

•p-p 
number  j  hence  — -  is  also  a  whole  number. 

Multiplying  the  second  equation  (2.)  by  B,  and  dividing 
by  P,  we  have 

P>_BRQ    BR^ 

■pin 

We  have  already  shown  that  -— .  produces  a  whole  num- 

ber ;  hence     ^  will  also  produce  a  whole  number.     This 

BR' 
being  the  case,  _— -  must  also  be  a  whole  number.     If  this 

operation  is  continued  till  the  number  which  multiplies  B 

R  V  1 
becomes  1,  we  shall  still  have  — — _,  equal  to  an  entire  num- 
ber 5  therefore  B  is  divisible  by  P. 


SECT,  v.]        REDUCTION  OF  FORMULAS.  12$ 

Hence,  if  a  number  will  exactly  divide  the  product  of  two 
numbers,  and  is  prime  with  one  of  them,  it  will  divide  the 
other. 

REDUCTION  OP  FORMULAS. 

214.  The  processes  of  generalization  which  we  have  no- 
ticed will  suggest  some  methods  of  demonstrating  formulas 
or  general  rules. 

FORMULAS    RELATING   TO    SIMPLE    INTEREST.  * 

215.  It  is  required  to  deduce  general  formulas  or  rules 
for  the  computations  relating  to  simple  interest. 

216.  To  present  the  subject  in  a  general  point  of  view,  let 
us  consider  the  five  things  that  enter  into  the  calculation, 
viz..  Principal^  Interest,  Rate,  Time,  and  Amount. 

Let  p=  principal,  t=  interest,  r=  rate  per  cent.,  t=  time, 
and  a=  amount. 

Taking  the  dollar  as  unity,  r  will  be  a  fraction,  whose  ae- 
nominator  is  100.  If  the  given  sum  be  put  at  interest  for 
one  year,  then  <=1  j  if  for  a  longer  period,  ^>1  ;  if  for  a 
shorter  period,  t^l.  The  interest  of  $1  will  evidently  be 
proportional  to  the  rate  and  time  jointly,  or  the  interest  of 
$l=rx^  The  rate  and  time  being  the  same,  any  given 
principal  will  be  to  any  other  principal  as  the  interest  of  the 
former  is  to  the  interest  of  the  latter. 

Hence  -        -         ^\  i  ^p  :  :rxt  :  i,  ox  i=pxrxt. 

By  making  the  necessary  transformations,  we  obtain  the 
four  following  formulas : 

1.         -        -  i=prt, 

2         -        -  p=iL 

^     rt 

3.     .     -      /=i. 

pr 


4. 


r=l 


pt 
These  formulas  may  be  enunciated  in  the  form  of  general 
rales. 


124  ELEMENTS    OF   ALGEBRA.  [sECT.  V. 

EXAMPLES. 

1.  What  is  the  interest  of  $3875,20  for  3  years,  at  7  per 
cent,  per  annum  % 

2.  What  is  the  interest  of  $325  for  3  months,  at  6  per 
cent,  per  annum  % 

3.  The  interest  of  a  certain  sum  is  $92,75,  the  time  3 
years  and  6  months,  and  the  rate  5  per  cent. :  what  is 
the  suml 

4.  The  interest  received  for  $4070,  at  9  per  cent.,  was 
$91,575  :  how  long  was  it  at  interest  1 

5.  The  sum  $367J  was  put  at  interest  for  6  months,  and 
at  the  end  of  the  time 'the  interest  paid  was  $183,55  • 
what  was  the  rate  per  cent.  % 

217.  Since  the  amount  is  equal  to  the  principal  +  the  in- 
terest, or  a=p-\-prt,  hy  making  the  necessary  transforma- 
tions we  shall  have 

1.         -        -        a=p-{-prt, 

%        -        '        p 


1+r? 

3.  -      ^      T=^i:2. 

pt 

4.  -        .         ^=^Zf. 

pr 

These  formulas  may  also  he  stated  in  the  form  of  gen- 
eral rules. 

EXAMPLES. 

1.  If;?=:$895,  r=7,  and  tz=z^,  what  is  the  value  of  a  ? 

2.  If  a=$7589,50,  r^S,  and  ^=5',  what  is  the  value  ofp^ 

3.  If  a=:$820,20,  ;)=:$600,  and  t—^,  what  is  the  numer- 
ical value  of  r  ? 

4.  If  a=$525,86,  ;}z=$35,80,    r=4,  what  is  the  numerical 
value  of  ^  ? 

FORMULAS   RELATING    TO    COMPOUND    INTEREST. 

218.  In  Compound  Interest  the  interest  is  supposed  to  re- 
main in  the  hands  of  the  borrower,  and,  being  added  to  the 


SECT,  v.]  REDUCTION    OF    FORMULAS.  1S5 

principal  at  the  end  of  each  year,  forms  a  part  of  the  princi- 
pal for  the  succeeding  year. 

219.  Let  p  and  r  sustain  the  same  relations  as  before,  and 
let  a=  the  amount  for  the  first  year,  and,  consequently,  the 
principal  for  the  second  year,  a'=:  the  amount  for  the  sec- 
ond year,  a"=  the  amount  for  the  third  year,  a"'=  the  amount 
for  the  fourth  year,  &c.  Then,  as  $1  will  be  to  any  given 
sum  as  the  amount  of  $1  for  one  year  is  to  the  amount  of 
that  given  sum  for  the  same  time,  we  shall  have 

\:p  ::  1+r  :  a,     or  a    =p(l-\-r), 

And  -  l:p(l-^r)  :  :  l+r  :  a',  or  a'  =Xl+r)^ 
And  -  1  :p{l-\-ry  :  :  1+r  :  a",  or  a"  =p(l+ry, 
And         ■         1  :pll-\-ry  :  :  1+r  :  a'",  or  a"  =p(l-{'r)\ 

&c. 
Let  A  represent  the  amount  for  n  years,  and  we  shall 
have 

1  :;)(l+r)'*-'  :  :  1+r  :  A,  or  A=p(l+ry. 

Hence,  by  making  the  necessary  transformations,  we  ob- 
tain the  following  formulas  :* 

1.  -         -        A=p(l+rY, 

2.  .        ^        p=-A.^. 

(1+rr 

These  formulas  may  be  stated  in  the  form  of  general  rules. 

EXAMPLES. 

1.  If  p  =  $3250,  71=8,  and  r=5,  what  is  the  numerical 
value  of  Al 

2.  If  A=$30,200,  n=20,  and  rz=6,  what  is  the  numerical 
value  of  ^  ? 

3.  If  A  =  $1479,15,  p=$1000,  and  n=6,  what  is  the  nu- 
merical value  of  r  ? 

♦  The  fourth  formula  is  omitted,  since  it  would  involve  Logarithms, 
which  are  treated  of  in  a  subsequent  section. 

Q2 


126  EI^BMENTS  OF  ALGEBRA.  [SECT.  V. 

FORMULAS  RELATING   TO   FELLOWSHIP. 

220.  Two  men  engage  in  trade  together,  and  furnish 
mopey  in  proportion  to  the  numbers  m  and  n  ;  they  gain  a 
sum  represented  by  g  j  it  is  required  to  deduce  formulas 
for  the  division  of  the  gain,  so  that  each  man  shall  receive 
his  equitable  share. 

Let  X—  the  share  of  the  first,  and  y—  the  share  of  the 
second : 

Then a?+y=g, 

And       -         -         -  a?  :  y  :  :  771  :  7i,  or  my—nx. 

Reducing  these  equations,  we  obtain 

x=J^^-. 


•-\-n 

Hence,  to  find  each  man's  share  of  the  gain.  Multiply  his 
stock  hy  the  whole  gain,  and  divide  the  product  by  the  whole 
stock  invested. 

Example.  Two  merchants,  A  and  B,  gained  by  trading  in 
company  $20,480.  A's  stock  was  $15,000,  B's  $18,000: 
what  was  each  man's  share  of  the  gain  % 

221.  Again:  suppose  three  persons  engage  in  trade,  and 
furnish  money  in  proportion  to  the  numbers  lUj  w,  and  p  ; 
they  gain  a  sum  represented  by  g  j  it  is  required  to  deduce 
formulas  for  the  division  of  the  gain  as  before. 

Let  a?,  y,  and  z  represent  the  respective  shares  of  the 
three  persons: 

Then  we  shall  have 

X  :  y  :  :  m  :  Uj  or  my—nx, 
X  :  z  :  :  m  :  Pj  OT  mz=px. 
Reducing  the  above  equations,  we  obtain 

x=_^ 

m-{-n-{-p' 


SECT,  v.]  REDUCTION    OF    FORMULAS.  127 

Hence,  to  find  each  man's  share  of  the  gain,  Multiply  his 
stock  by  the  whole  gain,  and  divide  the  product  by  the  whole 
amount  of  stock  invested. 

Example.  Three  merchants,  A,  B,  and  C,  gained  hy  tra- 
ding in  company  $1100}.  A's  stock  was  $1500,  B's  $1200, 
and  C*s  $850:  what  was  each  man's  share  of  the  gainl 

222.  As  the  above  formulas  contain  four  things,  viz^y 
whole  stock,  whole  gain,  the  particular  stock  whose  share  of 
the  gain  is  to  be  found,  and  that  share  of  the  gain,  it  is  ev- 
ident that  any  one  of  these  may  be  found  if  the  other  three 
be  given. 

Letting  S=  whole  stock,  S'=  stock  whose  share  of  the 
gain  is  to  be  found,  g=  the  whole  gain,  and  g'=  share 
of  the  gain  to  be  found,  and  substituting  these  letters 
in  the  preceding  formulas,  they  become 

1.  -        -        g'=^. 

^      S 

2.  .        -        g=^' 

.  S' 

3.  .        -       S'=^. 

g 

4.  .        .         S=^X 

8" 

EXAMPLES. 

1.  Two  men,  A  and  B,  traded  in  company,  with  a  joint 
capital  of  $8000;  they  gained  $1250.  A's  stock  was 
$3250 :  what  was  his  share  of  the  gain  1 

2.  Three  men.  A,  B,  and  C,  jointly  invest  in  trade  $2725  ; 
they  gain  $560,  of  which  A  receives  as  his  share  $120, 
6  receives  as  his  share  $160:  what  was  the  stock  in- 
vested by  each  1 

3.  Three  men,  A,  B,  and  C,  gain  by  trading  $6000.    A's 


12S  ELEMENTS  OF  ALGEBRA.  [sECT.  V. 

stock  was  $8000,  and  he  took  as  his  share  of  the  gain 
$2800 :  what  was  the  whole  stock  invested  1 
4.  Two  men,  A  and  B,  invest  in  trade  $3000.     A's  gain 
was  $250,  and  his  stock  $2600 :  what  was  the  whole 
gainl 

223.  Let  us  now  consider  the  cases  in  which  the  stock  of 
the  partners  in  trade  has  been  invested  for  different  lengths 
of  time. 

224.  Two  men  engage  in  trade  together,  and  furnish 
money  in  proportion  to  the  numbers  m  and  n,  for  the  times 
t  and  f  ;  they  gain  a  sum  represented  by-g:  it  is  required 
to  deduce  formulas  for  the  equitable  division  of  the  gain. 

Let  X  and  y  represent  the  respective  shares  of  the  gain : 
Then  we  shall  have 

nt'x^=zmty, 
Keducing  these  equations, 

mtor 

X=z^ 2__. 

y=      ^ 


mt-\-nt' 

These  results  may  be  enunciated  in  the  form  of  a  general 
rule. 

225.  Again  :  suppose  three  persons  invest  in  trade  money 
in  proportion  to  the  numbers  m^  tz,  and  p,  for  times  ^,  /',  and 
t'' ;  the  sum  gained  is  represented  by  g :  it  is  required  to 
deduce  formulas  for  the  equitable  division  of  the  gain. 

Let  a?,  y,  and  z  represent  their  respective  shares  of  the 

gain:  then 

x-\-yi-z=g, 

nt'x—mty, 

pt"x=mtz. 

Reducing  these  equations, 

mts: 
X=z o . 

mt-^nt'-\-pt" 


mt-^nt'-i-pt" 


SECT,  v.]  DISCUSSION.  129 

mt-\-TW -\-pt" 

These  results  may  also  be  enunciated  in  the  form  of  a 
general  rule. 

Example.  A,  B,  and  C  enter  into  partnership.  A  invests 
$1200  for  3  years,  B  $2000  for  2  years  and  9  months,  C 
$950  for  4  months.  They  gain  $2400  :  what  is  each  man's 
share  of  the  gain  ] 

DISCUSSION  OF  EaUATIONS  OF  THE  FIRST  DEGRER 

226.  When  a  question  has  been  solved  in  a  general  mai> 
ner,  that  is,  by  representing  the  known  quantities  by  letters, 
it  may  be  proposed  to  determine  what  values  the  unknown 
quantities  will  take  when  particular  suppositions  are  made 
upon  the  known  quantities.  This  is  called  the  discussion 
of  that  equation. 

227.  The  discussion  of  the  following  problem  presents 
nearly  all  the  circumstances  that  can  ever  occur  in  equa- 
tions of  the  first  degree.  * 

PROBLEM    OF    THE    COURIERS. 

A  courier  sent  out  from  a  certain  place  travels  in  a  right 
line  with  a  velocity  expressed  by  n.  After  the  first  courier 
had  travelled  a  distance,  a  second  was  despatched  after  him, 
travelling  with  a  velocity  expressed  by  m.  At  what  dis- 
tance from  the  starting  point  will  they  be  together  1 

In  order  to  render  the  conditions  of  the  question  more 
evident,  let  ED 


1 1 1 ( 1 

C  A  B  C 

represent  the  line  upon  which  the  couriers  travel,  A 
the  starting  point,  B  the  point  at  which  the  first  cou- 
rier is  when  the  second  starts,  and  C  the  point  at  which 
the  second  will  overtake  the  first : 

Let  x= AC,  and  y=BC : 

Then x — y=a^ 

R 


130  ELEMENTS    OF   ALGEBRA.  [sECT.  V. 

And —=y.. 

m    n 

Reducing  these  equations,  we  have 

am,         J  an 

-,  and  y: 


.^.  m — n  M — n 

DISCUSSION. 

I.  Let  m  >n. 

228.  In  this  case  the  values  of  x  and  y  will  be  positive, 
and  the  solution  of  the  problem  will  exactly  accord  with  the 
enunciation  j  for  if  the  second  courier  travels -faster  than  the 
first,  they  will  evidently  meet  somewhere  in  the  direction 
AD,  and  to  the  right  of  B. 

II.  Let  OT<  n. 

229.  In  this  case  the  values  of  x  and  y  will  be  negative. 
In  order  to  interpret  this  result,  we  observe  that,  the  courier 
from  B  travelling  faster  than  the  courier  from  A,  the  inter- 
val between  them  must  increase  continually.  It  is  absurd, 
therefore,  to  require  that  they  should  meet  in  the  direction 
AD.  The  negative  values  of  x  and  y,  then,  indicate  an  ab- 
surdity in  the  conditions  of  the  question.  To  remove  this 
absurdity,  we  have  only  to  suppose  that  the  two  couriers 
start  at  the  same  time  from  B  and  A,  and  travel  in  the  direc- 
tion BE',  in  which  case  the  equations  will  become 

y—x=za, 

And -=t 

m    n 

f^                                               am          J           an 
Or  -        -        -        0?— ,  and  y= , 

n — m  n—m 

which  give  the  values  of  x  and  y  positive,  and  indicate  that 
the  couriers  will  come  together  at  C  instead  of  C. 

III.  Let  m—n. 

230.  In  this  case  the  values  of  x  and  y  become 

am       am 

/». —  — 


m — n      0 

an    an 

m — n      0 


SECT,  v.]  THEORY    OF   NEGATIVE    QUANTITIES.  131 

In  order  to  interpret  this  result,  let  us  return  to  the  ques- 
tion. If  the  couriers  travel  with  equal  velocities,  it  is  ev- 
ident that  the  interval  between  them  must  always  continue 
the  same,  however  far  they  may  travel  in  either  direction. 
Indeed,  on  the  hypothesis  w=:w,  the  conditions  of  the  ques- 
tion produce 

And X — y=0, 

equations  which  are  incompatible  with  each  other.  It  is 
therefore  absurd  to  suppose  that  the  couriers  will  come  to- 
gether on  this  supposition. 

IV,  Let  m=n,  and  a=0. 
231.  In  this  case, 

Oxm     0 


X=:. 


m — n    m — n    0 
an    _Ox_m_0 
jn—n     m — n    0 
In  order  to  obtain  a  correct  interpretation  of  this  result, 
it  is  only  necessary  to  observe  that,  if  the  couriers  set  out 
each  from  the   same  point  at   the   same  time,  and  travel 
equally  fast,  there  is  no  particular  point  at  which  one  can 
be  said  to  overtake  the  other,  since  they  will  be  together, 
however  far,  and  in  whatever  direction   they  may  travel. 
Indeed,  on  this  supposition,  the  conditions  of  the  problem 
produce 

a? — y=0, 

x—y=0, 

two    dependant    or   identical    equations.     The    problem  is 

therefore   indeterminate,  since  we   have,   in  fact,  but   one 

equation  with  two  unknown  quantities. 

THEORY  OF  NEGATIVE  aUANTITIES. 
232.  It  has  already  been  shown,  • 

1.  That  adding  a  negative  quantity  is  the  same  as  sub- 
tracting an  equal  positive  quantity. 

2.  That  subtracting  a  negative  quantity  is  the  same  as 
adding  an  equal  positive. 


132  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 

3.  If  a  negative  quantity  be  multiplied  or  divided  by  a 
positive,  the  result  will  be  negative. 

4.  If  a  positive  quantity  be   multiplied  or  divided  by  a 
negative,  the  result  will  be  negative. 

5.  If  a  negative  quantity  be  multiplied  or  divided  by  a 
negative,  the  result  will  be  positive. 

233.  We  will  now  proceed  to  show  that,  if  the  conditions 
of  the  problem  are  such  as  to  render  the  unknown  quantity 
essentially  negative,  it  will  appear  in  the  result  with  the 
minus  sign,  although  it  may  have  been  regarded  as  positive 
in  the  statement  of  the  problem. 

1.  The  length  of  a  certain  field  is  a,  and  its  breadth  h: 
how  much  must  be  added  to  its  length  that  its  contents 
may  be  c  ? 
Let  xz=z  the  quantity  to  be  added  to  the  length: 
Then  a-{-x=i  whole  length. 

Since  the  area  of  a  field  is  found  by  multiplymg  its  length 
by  its  breadth,  we  have 

ab-\-lxz=:c. 

Reducing     -        -         -         xz=z- — a. 

0 

Now,  letting  a=8,  6=5,  and  0=60,  the  equation  becomes 

a:^^— 8  =  12— 8zzz4. 
5 

This  value  of  x  satisfies  the  conditions  of  the  problem  in 

the  precise  sense  in  which  it  was  stated. 

Again:  letting  a=:8,  6=5,  and  c=30, 

Then     ....        a:=r^— 8  =  6— 8=— 2. 

5 

In  order  to  interpret  this  negative  result,  let  us  return  to 
the  original  eq^uation : 

ab-\-lx—c. 

Substituting  -         -         -         8x5  +  5x— 2=30j 

Resolving  into  factors  -         -  5(8 — 2)  =  30. 

Hence  we  perceive  that,  though  addition  was  required  by 
the  enunciation,  yet  it  was  incompatible  with  the  conditions 


SECT.   ▼.]  EXPLANATION    OF    SYMBOLS.  133 

of  the  question  ;  and  the  algebraic  result,  true  to  the  condi- 
tions  of  the  question,  detects  the  error  in  the  enunciation, 
and  shows  that  x  is  to  be  subtracted  from  instead  of  being 
added  to  the  length  of  the  side.     Thus, 

ab — bxz=c, 

Or x=a-~^. 

o 
By  substitution       -        -        -        a?=8--^=8— 6=2. 

This  result  answers  to  the  question  modified  in  this  man- 
ner: 

The  length  of  a  certain  field  is  a,  and  its  breadth  b:  how 
much  must  be  subtracted  from  its  length  that  its  contents 
may  be  c  ? 

234-.  Discuss  in  like  manner  the  following  questions : 

2.  A  father  is  a  years  old,  and  his  son  b:  in  how  many 
years  will  the  son  be  one  fourth  as  old  as  the  father  1 

3.  A  man  when  he  was  married  was  a  years  old,  his  wife 
b :  how  many  years  before  his  marriage  was  he  t\vice 
as  old  as  she  1 

EXPLANATION  OF  SYMBOLS. 
Infinity. 
235.  A  mathematical  quantity  is  said  to  be  infinite  when 
it  is  supposed  to  be  increased  beyond  any  determinate  limits. 
The  symbols  usually  adopted  by  mathematicians  to  ex- 
press such  quantities  are  —  and  oo,  A  being  used  to  repre- 
sent any  finite  quantity. 

In  order  to  explain  these  symbols,  let  us  resume  the  equa- 
tions for  X  and  y  in  the  problem  of  the  couriers: 

am          1           an 
,andy=: ; 


ffi — n  tn — n 

Or,  if  m=n       -        -        x=—,      and  y=--. 

236.  In  order  to  explain  these  expressions  for  the  values 
12 


t34  ELEMENTS    OF    ALGEBRA.  [SECT.  V. 

of  X  and  y,  we  will  show  how  these  values  are  affected  by 
assuming  different  values  for  m  and  n. 
If  7w  =  3,  and  n—% 

3a,  and  y= =2o. 


3—2  "     3—2 

If  OT=:3,  and  %= 2,9, 

icz=?^30a,  and  w=?l?^=:29a. 

If»i=3,  and»=2,99, 

a;:zz^  =  300a,  and  y=?l^=299a. 
,01  ^       ,01 

IfOT=:3,  and7z=2,999, 

ic=-^=3000a,  and  y=M??i^=2999a. 
,001  '  ^       ,001 

If  »jzz:3,  and  71=2,9999, 

ir=-?^  =  30  000«,  and  y=M^^_?.=  29  999a,  &c. 
,0001  ^       ,0001  ' 

Hence  we  infer,  that  if  the  difference  between  m  and  n  be- 
comes less  than  any  assignable  quantity,  the  values  of  x  and 

y  will  be  greater  than  any  assignable  quantity ;  and  _  or  oo 

is  the  proper  symbol  of  infinity. 

237.  CoROL.  Since,  by  the  conditions  of  the  question,  a?= 
y+a,  which  will  continue  to  be  the  case  when  the  values  of 
X  and  y  become  infinite,  we  infer  that  one  mathematical  in- 
finity may  be  greater  thdn  another. 

Infinitesimal. 
239.  A  mathematical  quantity  is  called  an  infinitesimal, 
or  sometimes  nothing,  when  it  is  supposed  to  be  decreased 
below  any  determinate  limits. 

A 

The  symbols  used  to  express  such  a  quantity  are  —  or  0. 

CO 

In  order  to  explain  these  symbols,  let  us  resume  again  the 

equations 

am  J  an 

iC== ,  and  y=- 


w, — n  in — n 


SECT,  v.]  EXPLANATION    OF    SYMBOLS.  135 

239.  A  course  of  reasoning  similar  to  that  adopted  in  the 
preceding  case  will  show  that  the  values  of  x  and  y  decrease 
as  the  difference  between  m  and  n  increases.  Hence,  when 
that  difference  becomes  greater  than  any  assignable  quan- 
tity, the  values  of  x  and  y  become  less  than  any  assignable 
quantity. 

That  is,  x= = — ,  or  the  value  of  x  may  be  expressed 

m — n     00 

by  the  symbol  —  or  0. 

QO 

And  y=— —  =^,  or  the  value  of  y  may  be  expressed  by 
m — n      00 

the  symbol  _  or  0. 

QO 

240.  CoROL.  Since  x=y4-fl,  we  infer  that  one  infinitesimal 
may  be  greater  than  another. 

EXPLANATION    OF    THE    SYMBOL   OF   INDETERMINATION. 
0 
0* 

241.  A  quantity  is  said  to  be  indeterminate  when  every 
possible  value  will  satisfy  the  conditions  of  the  question. 

242.  The  symbol  used  to  express  indetermination  is  -. 

We  have  already  seen  that  the  equations  x=  "^  ,  and 

m — n 

,  on  the  hypothesis  m  —  n  and  a=0,  reduce  to  x= 


I — n 


-,  and  y=-y  and  also  that  all  possible  values  of  x  and  y  will 

satisfy  the  conditions  of  these  two  equations. 

243.  We  will,  however,  add  another  illustration  to  this 
case. 

I «» 

Take  the  expression :  if  we  perform  the  division,  the 

1 — X 

quotient  will  be  1  j  ahd  if  we  make  x=  1,  there  will  result 

U        .        .        .        !=£=!=?. 
l—x  0 


136  ELEMENTS    OF   ALGEBRA.  [sECT.  V. 

2.     -     -     -     h:^=i+x=i=l 

l—x  0 

3-   -   -   -   1eJ=i+-+-=3=5. 

drc,  ad  infin. 
Hence  every  possible  value  will  satisfy  the  conditions  — 

244.  It  should,  however,  be  observed,  that  this  symbol 
does  not  always  imply  indetermination. 

Thus,  the  expression  x— ,  if  a=b,  will  become 

or — r 

But,  resolving  the  terms  of  the  fraction  into  factors, 

^^(a-b){a'  +  ah-\-b')_d'-\-ab-\-b^ 

(a—bXa+b)      ~      a-\-b     '       ^        * 
which,  on  the  supposition  a—b^  becomes 


X  —  . 


a-\-a  '■la      2 


Hence  we  conclude  that  the  symbol  - ,  in  Algebra,  some- 
times indicates  the  existence  of  a  factor  common  to  the  two 
terms  of  the  fraction,  which,  in  consequence  of  a  particular 
hypothesis,  becomes  0,  and  reduces  the  fraction  to  the  form 

INEaUATIONS  OR  INEaUALITIES. 

245.  The  principles  established  respecting  equations  will 
in  most  cases  also  apply  to  inequations.  As  there  are  some 
exceptions,  we  will  here  illustrate  the  principal  transforma- 
tions which  may  be  made  upon  inequations,  and  then  apply 
those  transformations  to  the  determination  of  the  limits  of 
unknown  quantities. 

246.  Two  inequations  are  said  to  subsist  in  the  same  sense 
when  the  greater  quantity  stands  at  the  right  in  both  or  at 

*  See  Note  C. 


SECT,  v.]  INEQUATIONS.  137 

the  left  in  both,  and  ia  a  contrary  sense  when  the  greater 
quantity  stands  at  the  right  in  one  and  at  the  left  in  another. 

24-7.  1.  We  may  always  add  the  same  quantity  to  both 
members  of  an  inequality,  or  subtract  the  same  quantity 
from  both  members,  and  the  inequahty  will  continue  in  the 
same  sense. 

Thus,  let  -        -  2<12  ;  adding  6  to  both  sides, 

we  have         -         6+2<  124-6, 

Or     -        .        -  8<18. 

Again:  let         -  — 2>  — 12: 

Then         -        -         6— 2>6— 12,  or  4>— 6. 

CoROL.  A  term  may  be  transposed  from  one  member  of  an 
inequation  to  the  other. 

24-8.  2.  If  we  add  the  corresponding  members  of  two  or 
more  inequations  subsisting  in  the  same  sense,  the  inequa- 
tion which  results  will  exist  in  the  same  sense  of  those  added. 

Thus        -        -         -         8>5, 

And  -        -        -       10>2. 

Adding     -        -         19  +  8>2+5,  or  18>7. 

But  if  we  subtract  the  corresponding  members  of  one  in- 
equation from  another  subsisting  in  the  same  sense,  the  re- 
sulting inequation  will  not  always  exist  in  the  same  sense. 

Thus,  from        -        -        -        4.<7 

Subtract    -        -        -        -        2<3. 

There  will  remain      -  4— 2<7— 3,  or  2<4.. 

But  if  from        -        -         -         9<10 

We  subtract       -         -        -         6<  8, 

There  will  remain      -      •  -  9— 6>10— 8,  or  3>2. 

249.  3.  If  both  members  of  an  inequation  be  multiplied 
or  divided  by  any  positive  whole  number,  the  resulting  ine- 
quation will  exist  in  the  same  sense  as  the  inequation  mul- 
tiplied. 

Thus         -        -        -        -  6<10; 

Muhiplying  by  3        -         -         18<30.  ; 

Or,  again  ...         -  ^<^; 

Multiplying  by  6        -        -  2<3. 


138  ELEMENTS    OF    ALGEBRA.  [sECT.  T. 

CoROL.  An  inequation  may  be  freed  from  fractions  in  the 
same  manner  as  an  equation. 

250.  4.  If  we  multiply  or  divide  the  two  members  of  an 
inequation  by  a  negative  quantity,  the  resulting  inequation 
will  subsist  in  a  contrary  sense. 

Thus 6<10; 

Muhiplying  by  —3  -         -         -     — 18>— 30. 

Or,  again  ...         -  i<i5 

Multiplying  by  —6  -         -         -       — 2>— 3. 

Hence  it  also  follows,  that  if  we  change  the  sign  of  each 
term  of  an  inequation,  the  inequation  which  results  will  ex- 
ist in  a  sense  contrary  to  the  inequation  proposed  j  for  this 
.transformation  will  be  equivalent  to  multiplying  the  inequa- 
tion by  — !► 

EXAMPLES. 

1.  Find  the  limit  of  the  value  of  x  in  the  inequation 

„       23^  2a?  ,  . 
3^3 
Clearing  of  fractions,       21a:— 23>2a?+ 15  ; 
Transposing     -         -     '21a?— 2a?>  15  +  23; 
Reducing  -         -  19a?>38; 

Dividing  by  nineteen,  a?>2. 

2.  Find  the  limits  pf  the  value  of  x  in  the  inequations 

14a:+A>i|+230, 

7  0 

J^Tote. — To  determine  both  the  limits  of  a?,  it  is  necessary 
that  we  have  two  inequations  existing  in  a  contrary  sense. 
These  inequations  are  not  combined  together  like  equations, 
but  reduced  separately. 

3.  Find  the  limits  of  x  in  the  inequations 

X     X  ^7     2a? 
5"^¥>5"^'3'' 

X X      6 X 

7     14     5     10* 


SECT.  V  ]  INEQUATIONS.  139 

4.  The  double  of  a  number  diminished  by  5  is  greater 
than  25,  and  triple  the  number  diminished  by  7  is  less 
than  double  the  number  increased  by  13.  Required  a 
number  that  will  satisfy  the  conditions. 

Let  x=  the  number  :  then,  by  the  question,  we  have 
2x— 5>25, 
3x— 7<2a:+13. 
Resolving  these  inequalities,  we  have  ar>15  and  a:<20. 
Any  number,  therefore,  either  entire  or  fractional,  compri- 
sed between  15  and  *20,  will  satisfy  the  conditions. 

5.  A  shepherd  being  asked  the  number  of  his  sheep,  re- 
plied that  double  their  number  diminished  by  7  is  great- 
er than  29,  and  triple  their  number  diminished  by  5  is 
less  than  double  their  number  increased  by  16.  Requi- 
red a  number  that  will  satisfy  the  conditions. 

Resolving  the  question,  we  have  a:>18  and  x<21.  Here 
all  the  numbers  comprised  between  18  and  21  will  satisfy 
the  inequalities ;  but  since  the  nature  of  the  question  re- 
quires that  the  answer  should  be  an  entire  number,  the  num- 
ber of  solutions  is  limited  to  two,  viz.,  x=19,  07=20. 

6.  A  market-woman  has  a  number  of  oranges,  such  that 
triple  the  number  increased  by  2  exceeds  double  the 
number  increased  by  61,  and  5  times  the  number  di- 
minished by  70  is  less  than  four  times  the  number  di- 
minished by  9.  Required  a  number  that  will  satisfy  the 
conditions.  • 

7.  The  sum  of  two  numbers  is  32 ;  and  if  the  greater  be 
divided  by  the  less,  the  quotient  will  be  less  than  5,  but 
greater  than  2.     What  are  the  numbers! 

8.  A  boy  being  asked  how  many  apples  he  had  in  his 
basket,  replied,  that  the  sum  of  three  times  the  number 
plus  half  the  number  diminished  by  5,  is  greater  than 
16 ;  and  twice  the  number  diminished  by  one  third  of 
the  number  plus  2,  is  less  than  22.  Required  the  num- 
bers that  will  satisfy  these  conditions. 


140  ELEMENTS    OF    ALGEBRA.  [sECT.  V. 


SECTION  VI.  ; 

Involution  and  Powers, — Of  Monomials. — Of  Polynomials. — 
Binomial  Theorem. — Evolution  and  Roots. — Square  Root  of 
Jf umbers. — Cuhe  Root  of  JVumbers. — General  Method  of  ob- 
taining any  Root  of  Jfumbers. — Evolution  of  Monomials, — 
Of  Polynomials. — Calculus  of  Radicals. 

INVOLUTION  AND  POWERS. 

251.  Involution  is  the  multiplying  anumber  by  itself  till 
it  has  been  used  as  a  factor  as  many  times  as  there  are  units 
in  the  exponent. 

252.  The  product  thus  produced  is  called  the  power  of 
that  quantity ;  and  the  power  is  designated  ^r^if,  second^  third^ 
fourth^  &c.,  accordingly  as  the  number  has  been  used  once, 
twice,  three  times,  four  times,  &c.,  as  a  factor. 

253.  To  indicate  the  involution  of  a  polynomial,  or  of  a 
monomial  composed  of  several  factors,  the  numbers  should 
be  placed  within  a  parenthesis,  to  the  right  of  which  the  ex- 
ponent should  be  written. 

INVOLUTION  OF  MONOMIALS. 

254.  In  order  to  obtain  a  general  rule  for  the  involution 
of  monomials,  let  the  following  proposition  be  demonstra- 
ted, viz. :  The  power  of  the  product  of  two  or  more  factors 
is  equal  to  the  product  of  their  powers. 

Let  {abf  represent  the  second  power  of  the  product  of 

two  factors, 
And  a^lf^  the  product  of  the  second  power  of  the  same  fac- 
tors : 
Then  (abY—a^¥  ;  for,  by  the  definition  of  involution  (Art. 
251),  {abf=zabxab=aaxbb  =  a'b\ 

Again:  {abY—d^b'^}  for  {abY—obxabxabx  ....  taken 
m  times=aaa  .  .  m  iimesxbbb  .  .  m  times=a"'&"'. 


SECT.  VI.]  INVOLUTION    OF    MONOMIALS.  141 

255.  Now  let  it  be  required  to  involve  Sah^  to  the  fourth 
power : 

(3aby = 3ai«  x  3ab^  X3a6'x3aft«  =  3x3x3x3x  aaaa  x  bl'b^il'b* 
=  81a*6'=3*xa'^*x6'^*. 

256.  The  same  reasoning  will  evidently  apply  to  every 
case  of  monomials ;  hence,  for  the  involution  of  monomials 
we  have  the  following  general 

RULE. 

1.  Involve  the  coefficient  to  the  required  power. 

2.  Multiply  the  exponent  of  each  letter  by  the  exponent  which 
denotes  the  power  to  which  the  monomial  is  to  be  involved. 

J^ote  1. — If  the  number  to  be  involved  is  positive,  all  its 
powers  will  be  positive  (Art.  89).  If  it  be  negative,  the  even 
powers  will  be  positive  and  the  odd  powers  negative  (Art. 
90). 

^ote  2. — If  the  given  number  be  fractional,  involve  both 
the  numerator  and  denominator.  This  results  from  the  prin- 
ciple that  the  product  of  fractions  is  equal  to  the  product  of 
their  numerators  divided  by  the  product  of  their  denomina- 
tors (Art.  149).  Thus,  ('^y=?x-=^. 

Jfote  3. — The  above  rule  is  applicable  to  numbers  having 
negative  exponents,  since  the  negative  exponent  expresses 
the  reciprocal  of  a  power  (Art.  61).     Thus,  (a~^)^=a~'^^= 

JVb^e  4. — The  fourth  power  of  a  number  is  equal  to  the 
square  of  the  second  power  ;  thus,  (a)*=o  X  a  X a  x  a=oa  x  aa 
=(a')'.  The  sixth  power  is  equal  to  the  cube  of  the  second 
power  J  thus,  a^z=axaxaxaxaxa=aaxaaxaa=(a^f,  &;c. 

EXAMPLES. 

1.  Required  the  second  power  of  8a'6'.  •^tis,  640"^". 

2.  Required  the  third  power  of  5x^z,  Ans.  125jV. 

3.  Required  the  third  power  of  <6dfx,    Ans.  2l6a'yr'. 

4.  Required  the  fourth  power  of  2aVc*.    Ans,  lea'd'^c'*. 


142  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 

5.  Eequired  the  fifth  power  of  2a5V.        Ans.  32a^6'^a?2^ 

6.  Required  the  second  power  of  — 6a^Z>^     Jins.  36a^6'^ 

7.  Required  the  third  power  of  — 3aJc^. 

Ans.  — 27a%«. 

8.  Required  the  sixth  power  of  \d^h.  Ans.  i^-^-^c^^W, 

9.  Required  the  seventh  power  of  — 2a7^y. 

Ans.  -— 128a;'y. 

10.  Required  the  fourth  power  of  — 4^a^2>. 

Ans.  256a«6^ 

11.  Required  the  fourth  power  of — . 

Ans.     '''' 


12.  Required  the  second  power  of  3a~^.  Ans.  9a-*. 

13.  Required  the  second  power  of  1?^.     Ans.  _5?^*-. 

U.  Required  the  second  power  of  ??-.       Ans   ^^^^^ 


82a?  6724a?2 

15.  Required  the  third  power  of  6a~^b~\ 

Ans.  n6a-'b-^. 

16.  Required  the  fourth  power  of  Sah~^ 

'    Ans.  4>0%a'b-'^ 

17.  Required  the  fourth  power  of  lOx^z"^. 

Ans   10000a?'V*». 

18.  Required  the  fifth  power  of  ^a^xy^. 

Ans.  1024a^V2/'°. 

19.  Required  the  fifth  power  of  — Sabxy. 

Ans.  —U^a'b'xY. 

20.  Required  the  eighth  power  of  ba^x.-^ 

Ans.  390625a'V. 

21.  Required  the  fourth  power  of  ^.      Ans.  i5??^'. 

xyz  x^y^z^ 

22.  Required  the  second  power  of 


ISyz 

Ans, 


400j?^ 
324yV' 


ft 


SECT.  VI.]  INVOLUTION    OF    POLYNOMIALS.  143 


23.  Required  the  third  power  of 


We'd-* 
,1ns.      27a-*'^'^ 


24.  Required  the  fifth  power  of  Wb^c*dK 

Ans,  1024a«'J'»c»(i». 

25.  Required  the  seventh  power  of  — 2aa:ccP. 

Ans.  --128aVc'(/'\ 

26.  Required  the  nth  power  of  _^.         Ans.  ^  ^  .  -. 

^  ^  8cdy  S"c"d'Y 

27.  Required  the  eighth  power  of  —Wb^. 

Ans.  65536a'«6^. 

28.  Required  the  fourth  power  of  lOx^fz. 

Ans.  lOOOOa^'y^^. 

29.  Required  the  sixth  power  of  — 3a^b^c*d-\ 

Ans.  7•29a'«i'»c"(^*. 

30.  Required  the  third  power  of  12a-^b-^c-^dr*. 

Ans.  1728a-*6-»c-«(f-'«. 

INVOLUTION  OF  POLYNOMIALS. 
257.  Multiply  the  polynomial  by  itself  till  it  has  been  used 
as  a  factor  as  many  times  as  there  are  units  in  the  exponent  de- 
noting the  power  to  which  it  is  to  be  raised;  the  final  product 
will  be  the  power  required. 

EXAMPLES. 

1.  Required  the  first,  second,  third,  fourth,  and  fifth  pow- 
ers of  the  binomial  a-\-b, 
(a-\-by=a  -i-b      ------  1st  power. 


(a-hJ)*=a'+2a*+6' 2d  power. 

a+b 


a*+2a'6-h  ah" 


144 

(a+6) 


ELEMENTS    OF    ALGEBRA. 


a+b 

a'-j-3a'b-\-3aW-\-  ah'' 

a!'b  +  3a'b^-\-3ah'+b* 
(a-\-byz=a'-{-U'b-\-6a%^-^4>ah^-\-b^ 
a-\-b 
a'-{-4^a'b-j-   6a'b'-i-  4<a'b^-\-  a¥ 

{(l-\-hy=a'-^ba'b+  \Oa?b^+  lOa'b^-{-bab^-{-¥    - 

2.  Required  the  first,  second,'  third,  fourth, 
ers  of  the  binomial  a — h. 
{a — by=za — b      -         -         -         -         -'       - 
a — b 


c^ —  ab 


ab-\-W 


(^a-.bf=a^—^ah-{-b^ 
a  — b 


—  a^-{-2ab'—b' 
(a-^by=a^-3a'b+  3ab^—b^ 
a — b 


—  a'b-{-3a^b^—3ab''+b^ 


(a-'by=a'—Wb-\-Qa^W—^ab^+b'- 
a — b 


a^—^a'b^   6a'b' 
—  a'b-\-   Wb^ 


4>a'b^+  ab'      . 
Qa'h^+^ab'-l^ 


(a—by=a'—^a'b^-  IQa^b'^—lOa'b^+bah'—b' 

3.  Required  the  second  power  oi  a-\-b, 

<  a  -\-b 

a-\-b 


o^+  ab 

+  ab+b^ 
a^-^2ab-\-y,     Ans, 


[sect.  vi. 
3d  power. 


4th  power. 


5th  power, 
and  fifth  pow- 

1st  power. 


2d  power. 


3d  power. 


4th  power. 


5th  power. 


SECT.  VI.]  INVOLUTION    OF    POLYNOMIALS.  145 

J^ote. — Since  a  and  b  may  represent  any  numbers  what- 
ever, we  infer  the  following  general  principle  :  The  square 
of  a  binomial  is  the  square  of  the  first  term,  plus  twice  the 
product  of  the  two  terms,  plus  the  square  of  the  last  term. 
4.  Required  the  second  power  of  a—b. 
a—b 
a—b 


a"—  ab 
—  ab-\-b^ 

a^—2ab-\-b\     Ans, 

JhCott. — Since  a  and  b  may  represent  any  two  numbers,  a 
being  greater  than  6,  we  infer  the  following  general  princi- 
ple :  The  square  of  d  residual  is  the  square  of  the  first  term, 
minus  twice  the  product  of  the  two  terms,  plus  the  square 
of  the  last  term. 

5.  Required  the  second  pt)wer  of  6a4-3A. 

Ans.  36a'-|-36aJ-h95». 

6.  Required  the  second  power  of  la — 2J. 

Jlns.  49a'— 28a6+4J'. 

7.  Required  the  second  power  of  2a6+3c. 

Am.  4a'6'-fl2a&c+9c'. 

8.  Required  the  second  power  of  babe — 2acd, 

Ans.  25a'6V— 20a»*c'rf+4aVrf». 

9.  Required  the  third  power  of  2a -|- 36. 

Ans.  8a'+36a»6-f  54a^-f  276^ 

10.  Required  the  third  power  of  2a— 5&. 

Ans.  8a'— eOa'/j-hloOaft'— 1256*. 

11.  Required  the  second  power  of  a-|-l. 

^n*.  a«-f-2a-fl. 

12.  Required  the  second  power  of  2a — 1. 

wf;i«.  4a»— 4a-|-I. 

13.  Required  the  third  power  of  a+1. 

^n*.  a»+3a*+3a+l. 
13  T 


146  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 

3a— 1 


14f.  Required  the  second  power  of 


b+c 

Ans,  5^6.^1. 


15.  Required  the  third  power  of  a-\-b-\-c.       Ans,  (i?-\-3(i^'b 

+  3a'c4-  3a^>'+  3ac'+  6a&c+  Wc-i^il&-\-W-\-&, 

16.  Required  the  fourth  power  of  3a+26c. 

Ans.  81a^+216a='k+216a2iV+96aiV+  16&V. 

17.  Required  the  fifth  power  of  6a?— 2i.       ./?7i5.  7776a7^ — 

12960a?^Z>+8640:c^Z>2— 2880a?'^^+480a?&^— 32Z>^ 

18.  Required  the  second  power  of  6a4-2& — 3c. 

Ans,  36a'+  24''a6— 36ac+ 4.^>2— 126c+ Qc"; 

19.  Required  the  third  power  of  2a^ — 3x, 

Ans,  8a«— 36a^T+54aV— 27a:'. 
la—W 


20.  Required  the  second  power  of 


8a?+y 

*  64a:'-f  16a:y+/* 


21.  Required  the  second  power  of  — ± — 

Ans, 


9a?y 


9/^2— 246(i+  16(i^ 

6a~^b^ 
22.  Required  the  third  power  of 


Ans, 


3a— 1 

216a-'b' 


27a^— 27a^+9a— 1 

23.  Required  the  fourth  power  of  4a^6 — 2c^ 

.^715.  2b6a'^b'—51'ila'b'c^+3S4>a'b'c'—n8a'bc'-\-  16c\ 

24.  Required  the  fourth  power  of ^_. 

^^^   4>096aVy-^ 

^*'  ^H^aH^'+4^Tl' 
258.  Remark. — Any  factor  may  be  transferred  from  the 
numerator  to  the  denominator,  or  from  the  denominator  to 
the  numerator  of  a  fraction,  by  changing  the  sign  of  its  ex- 
ponent. 


SECT.  VI.]  BINOMIAL   THEOREM.  147 

1.  ?fl'^!!!xx-«=^xirArt.061)=4-. 

y     y  y    a?^  a^y 

'  2/~  2     y'      2     ^    ~     2    * 

CX-'    c     X-*    c  c      \      x^J     c  c 

.    2a-*    2    ^_,      1      2^1      ;,    2i» 
*•  ZT-i=^x^    Xr- 2=c:><-l><^=^-3• 
5.  ^=exx»=^^i=^-x-=^x4=-^. 
c      c  c  '  or     c  c    x-*    cx~* 

fi   <M?~*  _  ay* 

a-'b-'       1 


8. 


3a6       Sa^b'' 


Q    3a        _3a%* 

'  la-'b-y         T~" 

10 -^^  ,    =5a'Z>Vcf. 

BINOMIAL  THEOREM. 

259.  The  method  of  involving  polynomials  by  repeated 
multiplications  is  somewhat  tedious,  especially  when  high 
powers  arc  required.  This  has  led  mathematicians  to  seek 
for  other  methods.  The  most  simple  method  known  is  the 
one  invented  by  Sir  Isaac  Newton,  called  the  Binomial  The' 
orem.  Its  use  is  very  important  and  extensive  in  algebraic 
operations. 

260.  Let  us  take  the  binomird  a-h^,  of  which  a  is  called 
the  leading  quantity,  b  the  following  quantity.  Involving  by 
the  preceding  rule,  we  shall  find, 

(a-hiy=a'4-2a64-i* 

(a-f6)*=a'+3a«6-f   3a  6»4-^. 

(a+by=a'-\-Wb-^  6a^t^-\'  ^ab^-\-b*. 

(a+6)*=a*-f  5a*A-h  lOa^A^-f-  10a^6'-h  5a  b'+b\ 

la+by=a*+6a'b+  15a^A^4-20a'6'-f  15a^i*+   ^a  b'+b\ 

(cf+A)'=a'+7a«64-21a'6*H-35a«6*+35a'6*+21a**'+7aA«+3\ 


148  ELEMENTS    OF    ALGEBRA.  [sECT.  VI. 

261.  By  observing  the  several  results  above,  the  number 
of  terms  will  be  found  to  be  greater  by  1  than  the  index  de- 
noting the  power  to  which  the  binomial  is  to  be  expanded. 
Thus, 

The  square  has  three  terms  j 

The  cube  has  four  terms  ; 

The  fourth  power  has  five  terms ; 

The  fifth  power  has  six  terms  j 

The  sixth  power  has  seven  terms ; 

The  seventh  power  has  eight  terms ; 

And  if  the  nih.  power  of  a-\-b  were  required,  the  number 
of  terms  would  be  n-\- 1.  Hence,  if  the  index  of  a  binomial 
be  a  positive  whole  number,  the  number  of  terms  will  be 
one  greater  than  the  number  of  units  contained  in  the  index. 

262.  By  attending  to  the  exponents  of  the  letters  in  the 
above  powers,  we  shall  find  that  they  preserve  an  invariable 
order. 

In  the  square,  the  exponents  \  ^^  ^  ^^®  ^'  ■^'  ^  ' 

^  of  ^>  are  0,  1,  2. 

In  the  cube,  the  exponents  ^  of  «  are  3,  2,  1,  0  ; 
^  ^  of  5  are  0,  1,  2,  3. 

In  the  fourth  power,  the  exponents  \  ^^  ^  ^^®  ^'  ^'  2,  1,  0 ; 
^  "^  ^  of  Z>  are  0,  1,  2,  3,  4, 

&c. 

Two  laws  are  discoverable  here : 
•    1.  The  sums  of  the  exponents  of  the  two  letters  in  each 
term  are  equal,  and  each  sum  is  equal  to  the  index  de- 
noting the  power  to  which  the  binomial  was  to  be  raised. 

2.  The  exponent  of  the  leading  quantity  in  the  first  term 
is  the  same  as  the  index  denoting  the  power  to  which 
the  binomial  was  to  be  raised,  and  decreases  regularly 
by  1 J  the  exponent  of  the  following  quantity  is  1  in  the 
second  term,  and  increases  regularly  by  1. 

263.  If  it  be  required  to  involve  a-^b  to  the  power  denoted 
by  n^  the  exponents  of  a  would  be 

«,  n—l,  n—2j  n—Sy —3,  2,  1,  0  j 


SECT.  VI.]  BINOMIAL    THEOREM.  149 

Of  i,  0,  1,  2,  3, 71—3,  n—2y  w— 1,  n. 

Or,  expressing  the  letters  without  the  coefficients, 

b\ 

264.  The  same  principle  may  be  applied  if  the  exponents 
be  negative  or  fractional.     Thus, 

(a-f6)-*-o-»+a-='64-a-'^>'+a-'6'+a-«J'-far'5*4-,  &c.,  ad 

infin. 
Also, 

{a-\-b)^=J-^a~h-{-a~h'-{-a~^b^-\-a~ib'-ha~^!/'-{-,  &c.,  ad 
infin. 

It  is  evident  that  the  above  two  series  will  never  termi- 
nate, as  a  negative  or  fractional  index  can  never  become  0 
by  the  successive  subtractions  of  a  unit ;  hence,  when  the 
index  of  the  binomial  is  negative  or  fractional,  the  number 
of  terms  in  the  series  will  be  infinite. 

265.  The  law  of  the  coefficients  is  more  complicated,  though 
not  less  remarkable. 

In  the  preceding  series  of  powers  (Art.  259),  the  coeffi- 
cients taken  separately  are, 

-  -         1,     1. 

-  -      1,     2,     1. 
1,     3,     3,     1. 

1,    4.     6,    4,    1. 

-  1,    5,    10,  10,    5,    1. 
1,    6,  15,  20,  15,    6,    1. 

1,  7,  21,  35,  35,  21,  7,  1. 


In  the  first  power 
In  the  second  power 
In  the  third  power 
In  the  fourth  power 
In  the  fifth  power 
In  the  sixth  power 
In  the  seventh  power 
By  examining  the  above  series  of  coefficients,  it  will  be 
discovered, 

1.  That  the  coefficient  of  the  first  term  is  1. 

2.  That  the  coefficient  of  the  second  term  is  the  same  as 
the  index  denoting  the  power  to  which  the  binomial  is 
to  be  raised. 

3.  If  the  coefficient  of  any  term  be  multiplied  by  the  in- 
dex of  the  leading  quantity  in  the  same  term,  and  the 


150  ELEMENTS    OF    ALGEBRA.  [sECT;  VI. 

product  divided  by  the  index  of  the  following  quantity- 
increased  by  1,  the  quotient  will  be  the  coefficient  of 
the  following  term. 

266.  By  recurring  to  the  above  series  of  coefficients,  it 
will  be  observed  that  they  increase  and  then  decrease  in 
the  same  ratio,  so  that  the  coefficients  of  terms  equally  dis- 
tant from  the  first  and  last  terms  are  equal.  It  is  sufficient, 
then,  to  find  the  coefficients  oi  half  the  terms  ;  these,  repeat- 
ed in  the  inverse  order,  will  give  the  coefficients  for  the  re- 
maining terms. 

267.  By  inspecting  the  coefficients  farther,  we  shall  dis- 
cover that  in  any  power  of  a+i,  the  sum  of  the  coefficients 
is  equal  to  the  number  2  raised  to  that  power.  Thus,  the 
sum  of  the  coefficients 

In  the  second  power  is  -  J  -  -  4=: 2^; 

In  the  third  power  is      -  -  -  -  8==2^; 

In  the  fourth  power  is    -  -  -  -  16  =  2^; 

In  the  fifth  power  is        -  -  -  -  32=2' ; 

In  the  sixth  power  is      -  -  -  -  64  =  2^; 

In  the  seventh  power  i^  .  -  -  128=2', 

268.  If  it  be  required  to  involve  a-\-l  to  the  power  ex- 
pressed by  ra,  first,  taking  the  letters  and  exponents  without 
their  coefficients,  we  shall  have 

Let  A,  B,  C,^  (Sec,  represent  the  coefficients  of  the  several 
terms  in  order,  excepting  the  first  and  the  last,  which 
are  always  1. 

A=w,  coefficient  of  the  second  term. 

2 

B=^i:^,  coefficient  of  the  third  term. 

2    ' 

{n''-n){n-'X)  coefficient  of  the  fourth  term. 
2x3 
The  same  coefficients  may  be  used  in  the  inverse  order 
for  the  last  terms  of  the  indefinite  series.     Then  we  shall 
have,  by  restoring  the  coefficients, 


SECT.  VI.]  BINOMIAL    THEOREM.  151 

(fl-f  J)"=a"4-Aa'^'6+Ba'^'6»+Ca''-'6* .  .  .  Ca»6'»-»+Ba«J'^«-h 

269.  We  proceed,  in  the  next  place,  to  consider  the  signs 
to  be  prefixed  to  the  several  terms  produced  by  the  involu- 
tion of  a  binomial.  When  a  term  is  composed  of  several 
factors,  the  sign  of  the  term  will  evidently  depend  upon  the 
proper  signs  of  the  factors ;  if  an  even  number  of  them  be 
minus,  or  if  none  of  them  be  minus,  the  quantity  will  be 
positive ;  if  an  odd  number  of  them  be  minus,  the  quantity 
will  be  negative  (Art.  90).  Thus,  analyzing  the  fourth  pow- 
er of  a-{-by  each  term  is  composed  of  one  numerical  and 
four  literal  factors,  oil  plus j  and  consequently  each  term  will 
be  positive.     Thus, 

lXaXoXaxa=aVthe  first  term; 

^Xaxaxaxb=4^a^by  the  second  term; 

6Xaxaxbxb  =  6aW^  the  third  term  ; 

^Xaxbxbxb  =4.aA',  the  fourth  term  j 

lxbxbxbxb=b\  the  fifth  term. 
The  letters  and  exponents  are        a^-j-  a^b-^  a-b^-\-  ab*-\-b*; 
The  coefficients  are        -        -        1 +4«     +6      +4      +1. 

Compounding  the  series,  (a-f  6)S=a*  +  4ja'6H-6a*6*+4aZ>'-f  6*. 
Again,         (a—by=a*-Aa'b+ea^b^—4^ab^-{-b\ 

lxaxaXaxa=+a*,  the  first  term; 

4XaXaXax — b= — i-a'A,  the  second  term; 

6xaxax — bx — b=-{-6a^b^,  the  third  term; 

4xax — bx — kx — b= — 4a6^,  the  fourth  term; 

lx—b-\ bx—bx—b=  +  b\  the  fifth  term. 

The  letters  and  exponents  are        a*—  c^b-{-  a*b* —  ab'^+b*  ; 
The  coefficients  are         -        -        1+4     -f6       +4     +1. 

Compounding  the  series,  (a — by=a'^ — 4a'^-|-6a^6^ — ^ah'^+b*. 

270.  The  signs  of  the  terms  are  also  affected  by  the  sign 
of  the  exponent. 

Let  it  be  required  to  expand  (a+d)"*  i 


152  ELEMENTS    OF    ALGEBRA.  [SECT.  VI. 

The  letters  and  ^    ^.,^  ^.3^^  ^_,^,_^  a-^^^^  &c.,  ad  infin. ; 

exponents  are  S 
The  coefficients  )    j   _2       ^3       _^^         &c.,adinfin. 


\     ar'^ —  a~^b-{-  ar^W —  a~^Z>^,  &c.,  ad  infin. ; 


are 

Multiplying,        "" 

(a4-Z.)-2=ra-2— 2a-^^>+  Sa-^i^— 4a-^6^  &c.,  ad  infin. ; 

Ur,  (^+6)    =— — —A- — — — >  &c.,  ad  mtin. 

a^     a^      or      a' 

Again:  let  it  be  required  to  expand  (« — &)~^: 

The  letters  and 
exponents  are 
The  coefficients  ^     ^   _^       _^3    .   _^^         &c.,  ad  infin. 

are  J 

Multiplying, 

(a_  j)-2^a-2_f_2a-364-3a-^&2+4a-55^  &c.,  ad  infin. ; 

Or,  (a— ^)    — -,+-T+— r  +  -^j  ^c.,  ad  infin. 

a^     or      a*      a"" 

271.  The  principles  of  the  Binomial  Theorem  may  be 
stated  as  follows : 

I.  The  exponent  of  the  leading  quantity  in  the  first  term  of  the 
power  is  the  same  as  the  index  denoting  the  power  to  which  the 
binomial  is  to  be  raised^  and  decreases  regularly  by  1.  The  ex- 
ponent of  the  following  quantity  is  1  in  the  second  term,  and  in- 
creases regularly  by  1  in  the  succeeding  terms. 

II.  The  coefficient  of  the  first  term  is  1 ;  that  of  the  second  the 
same  as  the  power  to  which  the  binomial  is  to  be  raised ;  and 
universally^  if  the  coefficient  of  any  term  be  multiplied  by  the 
exponent  of  the  leading  quantity  in  that  term,  and  the  product  be 
divided  by  the  exponent  of  the  following  quantity  -j-1,  the  result 
will  be  the  coefficient  of  the  succeeding  term. 

JVote  1. — The  learner  will  find  it  convenient  to  obtain  the 
series  of  the  letters  and  exponents,  and  the  series  of  coeffi- 
cients separately,  and  then  compound  them  by  multiplying 
their  corresponding  terms,  as  in  the  preceding  cases. 

J^ote  2. — The  preceding  discussions  relating  to  the  Bino- 
mial Theorem  will  suggest  some  methods  of  verifying  the 
work,  and  also  of  abridging  it. 


SECT.  TI.]  BINOMIAL   THEOREM.  153 

EXAMPLES. 

1.  Required  the  fourth  power  of  a+b. 
Expanding  letters,  &c.,         a*-{-  a^b-\-  a'i'-f  ai^-\-b*', 
Finding  coefficients      -         1+^     -\-6      -}-4      +1. 

Compounding,  (a-{-bY=a*-\-Wb-^Qa'b^+^aly'i-b',     jJns 

2.  Required  the  fourth  power  of  a — b. 

The  letters  and  exponents  are     a* —  c£^b-\-  a^l^ —  aft* -f  5*5 
The  coefficients  are       -        -      1  -|-4     +6      +4      +1. 

Compounding,       -         {a—by=a*—^(^b^^a^b'—^a}^-\-b\ 

3.  Required  the  fifth  power  of  a-\-b  and  of  a — b. 

4.  Required  the  sixth  power  of  a+^  and  of  a — x. 

5.  Required  the  seventh  power  of  a:-fy  and  of  x — y. 

6.  Required  the  eighth  power  oi  a-\-b  and  of  a — b. 

7.  Required  the  eighth  power  of  x — y. 
8?  Required  the  fourth  power  of  \-\-a. 

Expanding  the  terms        -  l*-hl'xa+l'Xa*-fl'xa'4-o*; 

Finding  coefficients        -  1  +4        +6         4-4         +1. 

Compounding,  and  reject-  >  ^ 
mg  the  factor  1       -        J 

9.  Required  the  fourth  power  of  3a4-25. 

Let  x=3a,  and  y  =  25:  the«  (3a+26)*=(x-f  y)\ 
Expanding  this  last  expression,     x*+  or'y-f-  xy-|-  xy^-\-y^l 
Finding  coefficients       -        -       .1+4      -|-6       +4     -fl. 


Compounding        -  (a:+yy=x*-t-4ar'y-f-6a:'y*  +  4jry'-}-y*. 

Restoring  the  values  of  x  and  y, 

(3a  +  2Z.y  ^  {Zay  +  4  x  (Sa^  x  25  +6  x  {^af  x  (25)*4-4  X 
3ax(25)='-}-(25)*. 
Involving  the  terms, 

(3^+25)*  =  81a*  +  4x270^x25  +  6  X9a«x45»+4X  3ax 
85»+165*. 
Multiplying  factors, 

(3a+25)*  =  81a*+216a'54-216a'5»+96a5»+165*. 

10.  Required  the  fifth  power  of  2cx — 4y. 

Let  a=2cx,  and  5=4y  :  then  {lex — 4y)*=(a— 5)*. 

U 


154  ELEMENTS    OF    ALGEBRA.  [sECT.  VI. 

Expanding  the  terms,        a^—.a'b-\-     aW—     a%'^-\-ah''—b^i 
Finding  coefficients,  1+5      +10       +10      +5      +1. 

Compounding,     {a—Vf=za''—ba'h-\.l(ia?b''—10a%^+bab''—l\ 
Restoring  the  values  of  a  and  J, 

(2ca:-4y)^z=  32cV— 320cVy  +  1280cV/— 25600^/ + 
2560ca??/*— 1024^^ 

11.  Required  the  fourth  power  of  a^  ^b^. 

Let  x^a\  and  y=Z>^:  then  {a^+by={x-]-yy. 
Expanding  the  terms  -         a:'*+  a:^y+  a:^  +  x'if-\-y'^ : 

Finding  the  coefficients     -         1+4      +6       +4      +1. 

Compounding         -     (a?+2/)''— a?''+4ar'y+6a?y+4a?y'+y*. 
Restoring  the  values  of  x  and  y, 

{a^+by=a?-\-^a%^-\-Qa'b'+A>a^b^-\-VK 

12.  Expand  (2^^— 5J)^ 

13.  Expand  {Zabx+yy, 

14.  Expand L_z=(a+*)-^ 

{a+bf     ^   ^ 

15.  Expand  (a+a:)"^. 

16.  Expand  {a-^b)~K 

17.  Expand  (6a5c — laxyy. 

18.  Expand  (33?^— 4y)^ 

19.  Expand  {c^—Qax)K 

20.  Expand  (3a^— 1)*. 

272.  The   powers   of  any  polynomial  whatever   may  be 
found  by  the  Binomial  Theorem.     Take,  for  example,  {a-\-b 
+c)^     Letting  a?=:&+c,  we  shall  have 
{aJtb-{-cy={a-\-x)\ 
Expanding      -         -         -         (a+a?)^=a^+3a^a?+3aa?^+a7'. 
Restoring  the  value  of  a?, 

(a  +  Z>+cy=za^+3a^(^>  +  c)  +  3a(i  +  c)2+(&+c)^ 
Expanding  and  multiplying  factors, 

(a+Z>+cy=a'+3a'^>+3a2c+3a6'  +  6ak  +  3ac2+&=»+3J'c 
+  3k2+c^ 

2.  Required  the  third  power  of  a — b-\-c. 

3.  Required  the  third  power  of  25c — 3a;+y. 


SECT.  VI.]        EVOLUTION  AND  ROOTS.  165 

EVOLUTION  AND  ROOTS.       . 
Extraction  of  the  Square  Root  of  JVumhers, 

273.  A  power  of  a  number  has  already  been  defined  to  be 
the  result  of  multiplying  the  number  into  itself  continually, 
until  the  number  has  been  used  as  a  factor  as  many  times  as 
there  are  units  in  the  exponent  denoting  the  power. 

The  second  power  of  6:r6  ><6  =  36. 

The  third  power  or  cube  of  6::^6  x  6  x  6  =  216. 

The  fifth  power  of  47=47  x  47  x  47  x  47  x  47=229,345,007. 

Involution  is  the  method  of  finding  the  various  powers  of 
numbers. 

Evolution  is  the  reverse  of  this:  it  explains  the  method  of 
resolving  a  number  into  equal  factors,  called  roots. 

274.  When  a  number  is  resolved  into  two  equal  factors, 
one  of  the  factors  is  called  the  Square  Root ;  when  resolved 
into  three,  the  Cube  Root  j  when  into  four,  the  Fourth  Root, 
Sec. 

The  first  ten  numbers  are  1,2, 3,    4,    5,    6,    7,    8,    9,    10. 

And  their  squares      -       1,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

By  inspecting  this  table,  it  will  be  perceived  that  among 
entire  numbers,  consisting  of  one  or  two  figures,  there  are 
nine  only  which  are  squares  of  other  numbers.  The  square 
roots  of  other  numbers,  expressed  by  one  or  two  figures,  will 
be  found  between  two  whole  numbers  differing  from  each 
other  by  unity.  Thus,  55,  comprised  between  49  and  64, 
has  for  its  square  root  a  number  between  7  and  8 ;  78  has 
for  its  square  root  a  number  between  8  and  9.  The  num- 
bers in  the  second  line  of  the  table  being  the  squares  of 
those  in  the  first,  the  numbers  in  the  first  are  the  square 
roots  of  those  in  the  second ;  therefore  the  square  root  of 
numbers  consisting  of  one  or  two  figures  will  readily  be 
found  by  the  table. 

275.  Let  it  next  be  required  to  find  the  root  of  a  number 
consisting  of  more  than  two  figures.  It  has  already  been 
shown  that  the  square  of  any  binomial,  as  (a-\-by=za-\-2ab 
+6*.     Every  number  may  be  regarded  as  made  up  of  a  cer- 


156  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 

tain  number  of  tens  and  a  certain  number  of  units ;  thus, 
46  is  composed  of  4  tens  and  6  units,  and  may  be  expressed 
thus,  40  +  6;  the  square  of  which  may  be  obtained  in  the 
same  manner  as  the  square  of  a-{-b  ;  thus, 

40  +  6 

40  +  6 

1600+240 

240+36 


1600+480  +  36=^2116. 

In  this  result,  as  in  the  Square  of  the  binomial  a-{-b,  in 
which  a  may  represent  tens  and  b  units,  it  will  be  observed 
there  are  three  parts,  viz. :  the  square  of  the  tens,  40^,=  1600 ; 
twice  the  product  of  the  tens  by  the  units,  2x40x6=480; 
and  the  square  of  the  units,  6^=36.  These  three  parts  will 
be  found  in  the  second  power  of  every  number. 

276.  We  next  proceed  to  reverse  this  process,  and  find 
the  square  root  of  2116. 

As  the  square  of  4  tens,  or  40,  is  1600,  and  the  square  of 
5  tens,  or  50,  is  2500,  the  root  can  contain  only  4  tens. 
Subtracting  the  square  of  this         -         -         -         2116 

Square  of  4  tens,  or  40        -        -         -        -         1600 

516 
This  remainder  contains  twice  the  product  of  the  tens  by 
the  units,  plus  the  square  of  the  units.  Now,  if  we  double 
the  tens,  which  gives  80,  and  divide  516  by  80,  the  quotient 
is  the  figure  of  the  units,  or  a  figure  greater  than  the  units. 
This  quotient  figure  can  evidently  never  be  too  small,  but 
it  may  be  too  large,  as  516,  besides  containing  double  the 
product  of  the  tens  by  the  units,  may  contain  tens  arising 
from  the  square  of  the  units.  The  figure  representing  the 
units  can  never  be  greater  than  9.  516-^80=6.  To  ascer- 
tain whether  6  express  the  units,  we  multiply  80  by  6  =  480, 
and  subtract  it  from  516:  the  remainder  is  36;  from  this 
subtract  the  square  of  the  units  6  x  6  =  36 :  the  remainder  is 
0 ;  hence  4  tens  and  six  units,  or  46,  which  is  the  root. 


SECT.  VI.]  SQUARE    ROOT    OF    NUMBERS.  16t 

The  operation  will  stand  thus : 
2116^ 
1600 


2116^40-f-6=46,  root. 


40x2=80)516 
480 

""is 

6x6=         36 

0 

277.  The  work  may  be  abridged  by  several  modifications. 
By  observing  the  table  of  the  squares  of  the  numbers  1,  2, 
3,  4,  &c.,  it  will  be  perceived  that  the  square  of  a  num- 
ber consisting  of  one  figure  can  contain  no  figure  of  a  high- 
er denomination  than  tens.  If  we  annex  a  cipher  to  the 
numbers  1,  2,  3,  4,  &c.,  they  become 

10,    20,     30,      40,      50,      60,      70,.     80,      90,      100; 
And  their  squares  are 
100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000. 

From  which  we  see  that  the  square  of  tens  will  contain  no 
figure  of  a  less  denomination  than  hundreds,  nor  higher 
than  thousands.  When,  then,  the  square  root  of  a  number 
consisting  of  three  or  four  figures  is  required,  in  finding  the 
tens,  we  may  reject  the  first  two  figures  on  the  right,  as  they 
can  in  no  way  influence  the  result.  As  the  square  of  hun- 
dreds can  contain  no  figure  of  a  less  denomination  than 
thousands,  when  the  square  root  of  a  number  consisting  of 
five  or  six  figures  is  required,  in  obtaining  the  hundreds  we 
may  reject  four  figures  at  the  right  liand.  When,  then,  the 
square  root  of  any  number  is  required,  we  may  divide  it 
into  periods  of  two  figures  each  (if  a  number  consist  of  an 
odd  number  of  figures,  the  last  period  will  contain  but  one 
figure),  and  the  number  of  these  periods  will  be  the  number 
of  figures  in  the  root.  Each  of  these  periods,  in  connexion 
with  the  remainder  resulting  from  the  operations  on  the 
preceding  period,  may  be  used  independently  of  the  follow- 
ing periods  in  obtaining  that  figure  of  the  root  contained  in 
14 


158  ELEMENTS    OF    ALGEBRA.  [sECT.    VI. 

it.  In  the  above  example,  likewise,  in  which  the  square 
root  of  2116  is  required,  as  the  product  of  tens  by  units  ev- 
idently can  contain  no  figure  less  than  tens,  after  subtract- 
ing the  square  of  the  tens,  the  next  step,  the  division,  may 
be  as  well  performed  after  rejecting  the  cipher  from  the 
right  of  the  tens,  and  the  unit  figure  from  the  right  of  the 
dividend.  Moreover,  it  will  be  perceived  that,  instead  of 
finding  first  twice  the  product  of  the  tens  by  the  units,  and 
then  the  square  of  the  units,  we  may  obtain  the  sum  of  both 
numbers  by  placing  the  unit  figure  at  the  right  of  the  tens 
in  the  divisor,  and  multiplying  the  result  by  the  unit  figure. 
With  these  modifications,  the  work  of  extracting  the  square 
root  of  2116  will  stand  thus  : 

2il6]46 
16 


86)516 
516 
Find  the  square  of  the  tens  in  the  first  period  j  subtract, 
and  bring  down  to  the  right  of  the  remainder  the  next  peri- 
od. Divide  by  twice  the  tens,  rejecting  the  right-hand  fig- 
ure of  the  dividend.  Place  the  quotient  figure  in  the  root, 
and  at  the  right  of  the  divisor,  and  multiply  this  last  num- 
ber by  the  quotient  figure,  and  subtract  j  as  there  is  no  re- 
mainder, 46  is  the  root. 

Kequired  the  square  root  of  53361. 
5336i|231 
4 

^43)133 
129 


461)461 

461.     ^ns.  231. 
278.  The  same  process  may  be  extended  to  any  number, 
however  large.    From  the  preceding  operations,  the  follow- 
ing rule  for  the  extraction  of  the  second  root  will  be  readily 
inferred : 


SECT.  VI.]  SQUARE    ROOT    OF    NUMBERS.  159 


RULE. 


I.  Separate  the  number  into  periods  of  two  figures  each^  be- 
ginning  at  the  right  hand:  the  left  hand  period  mil  often  con- 
tain  but  one  figure, 

II.  Find  the  greatest  square  in  the  first  period  on  the  left ; 
write  the  root  in  the  place  of  a  quotient  in  division^  and  subtract 
the  second  power  from  the  left-hand  period. 

III.  Bring  down  the  next  period  to  the  right  of  the  remainder 
for  a  dividend^  and  double  the  root  already  found  for  a  divisor. 
See  how  many  times  the  divisor  is  contained  in  the  dividend,  ex* 
elusive  of  the  right-hand  figure,  and  place  the  result  in  the  root^ 
and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor  thus  augmented  by  the  last  figure  of 
the  root,  and  subtract  the  product  from  the  dividend,  and  to  the 
remainder  bring  down  the  next  period  for  a  new  dividend. 

V.  Douhle  the  whole  root  already  found  for  a  new  divisor ^ 
and  proceed  as  before,  till  all  the  periods  are  brought  down.  The 
root  will  be  doubled  if  the  right-hand  figure  of  the  last  divisor  be 
doubled. 

If  there  is  no  remainder  after  all  the  periods  are  brought 
down,  the  proposed  number  is  a  perfect  square.  If  there  is 
a  remainder,  by  the  above  rule,  the  root  of  the  greatest 
square  number  contained  {n  the  proposed  number  will  be 
obtained. 

When  the  proposed  number  is  not  a  perfect  square,  a 
doubt  may  arise  whether  the  root  found  be  that  of  the  great- 
est square  contained  in  the  number.  This  may  be  deter- 
mined by  the  following  rule.  The  square  of  a-|-l  is  a'-f-2a 
-f  1 J  whence  the  square  of  a  quantity  greater  by  unity  than 
a  exceeds  the  square  of  a  by  2a+l  ;  or,  the  difference  be- 
tween the  squares  of  two  consecutive  numbers  is  equal  to  twice 
the  less  number  augmented  by  unity. 

Hence  the  entire  part  of  the  root  cannot  be  augment- 
ed unless  the  remainder  exceed  twice  the  root  found  plus 
unity. 


160  ELEMENTS  OF  ALGEBRA.  [sECT.  VI. 

Required  the  square  root  of  1287135 

9     '~ 


65)387 
325 

Now,  as  35x2+l=71>62,  35  is  the  entire  part  of  the 
root. 

EXAMPLES. 

1.  What  is  the  square  root  of  451,5841  ^ns.  672. 

2.  What  is  the  square  root  of  9,186,9611       Jins.  3031. 

3.  What  is  the  square  root  of  13,032,1001     ^ns.  3610. 

4.  What  is  the  square  root  of  4,543,164,409  1 

^ns.  67,403. 

5.  What  is  the  square  root  of  669,420,148,761 1 

^ns.  818,181. 

279.  From  Avhat  has  been  done,  it  will  be  perceived  that 
there  are  many  numbers  the  roots  of  which  are  not  whole 
numbers;  and  although  there  must  be  a  number  which,  mul- 
tiplied into  itself,  will  produce  any  number  whatever,  yet 
these  numbers  can  have  no  assignable  roots,  either  among 
whole  or  fractional  numbers.  The  proof  of  this  depends  on 
the  following  proposition,  which  has  already  been  demon- 
strated (see  Art.  213) : 

Every  number,  P,  which  will  exactly  divide  the  product, 
AxB,  of  two  numbers,  and  which  is  prime  to  one  of  them, 
will  divide  the  other. 

The  root  of  an  imperfect  power  evidently  cannot  be  ex- 
pressed by  a  whole  number,  and,  to  show  that  it  cannot  be 
expressed  by  a  fraction,  let  c  be  an  imperfect  square  j  if  its 

root  can  be  expressed  by  a  fractional  number,  let  -  repre- 

a 

sent  that  fractional  number :  then  we  shall  have  |^ 

o 
Or      ...        -        c=_. 


SECT.  VI.]  SQUARE    ROOT    OF    NUMBERS.  161 

If  c  be  not  a  perfect  square,  its  root  will  not  be  an  entire 
number  j  that  is,  a  will  not  be  divisible  by  b  ;  but  it  has  been 
demonstrated  that  if  a  is  not  divisible  by  Z>,  axa  or  a'  is  not 

2 

divisible  by  b  or  bxb=b'^,  whence  —  cannot  be  equal  to  an 

tr 

entire  number  c. 

All  numbers,  both  entire  and  fractional,  have  a  common 
measure  with  unity ;  on  this  account  they  are  said  to  be 
commensurable  j  and  since  the  ratio  of  these  numbers  to 
unity  may  always  be  expressed,  they  are  called  rational 
numbers. 

The  root  of  a  numbeir  not  a  perfect  square  can  have  no 
common  measure  with  unity,  as  no  fraction  can  be  assigned 
sufficiently  small  to  measure  at  the  same  time  this  root  and 
unity.  The  roots  of  such  numbers  are  called  incommensu- 
rable or  irrational  numbers.     They  are  likewise  called  surds. 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS. 

280.  The  square  root  of  a  fraction  may  be  found  by  ex- 
tracting the  square  root  of  the  numerator  and  of  the  denom- 
inator ;  thus,  the  square  root  of  ^s  is  |.  If  the  numerator 
or  denominator  is  not  a  perfect  square,  the  root  of  the  frac- 
tion cannot  be  found  exactly,  but  the  root  to  within  less 
than  one  of  the  equal  parts  of  the  fraction  may  readily  be 
found  by  the  following 

RULE. 

Multiply  both  terms  of  the  fraction  by  the  denominator  which 
does  not  change  the  value  of  the  fraction  ;  then  extract  the  square 
root  of  the  perfect  square  nearest  the  value  of  the  numerator^  and 
place  the  root  of  the  denominator  under  it  ;  this  fraction  mill  be 
the  approximate  root. 

Required  the  square  root  of  | :  multiply  both  terms  by  5, 
which  gives  ^^y  of  which  f  is  the  required  root  exact  to 
within  less  than  |.  We  might  multiply  both  terms  of  if  by 
any  perfect  square,  and  thus  approximate  the  root  mor« 
nearly.    Thus,  multiplying  by  144,  it  becomes  ||^§,  the  rotti 

X 


162  ELEMENTS    OF    ALGEBRA.  [sECT.  VI. 

of  which  is  nearest  ||.  Thus  we  have  the  root  of  f  to  within 
less  than  j^^. 

The  approximate  root  of  a  number  not  a  perfect  square 
may  be  found  in  a  similar  manner  within  a  given  fraction. 

Multiply  the  proposed  number  by  the  square  of  the  denomina' 
tor  of  the  fraction  ;  then  extract  the  square  root  of  the  product  to 
the  nearest  unit,  and  divide  this  root  by  the  denominator  of  the 
fraction. 

This  rule  may  be  demonstrated  as  follows : 

Let  a  be  the  number  proposed,  of  which  it  is  required  to 

find  the  root'to  within  less  than  -  :  a—-—  ;  let  r  be  the  en- 

n  TV- 

tire  part  of  the  root  of  the  numerator  an^ ;  ar^  will  be  com- 
prised between  r^  and  (r-|- 1)^ ;  consequently,  the  square  root 

of  a  will  be  comprised  between  those  of  —  and  1_2_jl  that 

r^  n^ 

is,  between  -  and  ^        ^,  whence  -  will  be  the  root  of  a  to 
n  n  n 

within  less  than  -. 
n 

Find  the  square  root  of  59  to  within  less  than  ■^^' 
59'x  (12)^=8496. 

V  8496  =  92.     ff,  ^ns, 

281.  The  manner  of  determining  the  approximate  root  in 
decimals  is  a  consequence  of  the  preceding  rule. 

To  obtain  the  square  root  of  a  number  within  J^,  yi^, 
to-Vtt?  ^^">  "^6  multiply,  by  the  preceding  rule,  the  number 
by  (10)^,  (100/,  &c.,  or,  what  is  the  same  thing,  we  add  to 
the  right  of  the  number  two,  four,  six,  &c.,  ciphers  j  then 
extract  the  square  root  of  the  product  to  the  nearest  unit, 
and  divide  this  root  by  10,  100,  1000,  &:c. 

The  number  of  ciphers  annexed  to  the  whole  number 
should  always  be  double  the  number  of  decimal  places  re- 
quired to  be  found  in  the  root.  The  roots  of  decimal  frac- 
tions, whole  numbers,  and  decimals,  may  be  found  by  the 
preceding  rules.     The  number  of  decimals  in  the  proposed 


8£CT.  YI.]  CUBE    ROOT    OF    NUMBERS.  163 

number  must  always  be  made  even  by  annexing  ciphers  if 
necessary.  A  vulgar  fraction  may  be  changed  to  a  decimal 
fraction  before  extracting  its  root,  and  a  mixed  number  to 
a  whole  number  and  decimal. 

EXAMPLES. 

1.  What  is  the  square  root  of  31,027  to  within  ,01 1 

Ans.  5,57. 

2.  What  is  the  square  root  of  0,0100,1  to  within  ,00001  \ 

Ans,  0,10004. 

3.  What  is  the  square  root  of  \\  to  withiti  ,001  \  \ 

Ans.  0,886.  * 

4.  What  is  the  square  root  of  2}^  to  within  0,0001 1 

Ans.  1,6931. 

5.  What  is  the  square  root  of  7  *?  Aiis.  2,645+. 

6.  What  is  the  square  root  of  4-^  1  Ans.  2,027+. 

7.  What  is  the  square  root  of  i^  1  Ans.  0,8044+. 

8.  What  is  the  square  root  of  0,01001 1 

Ans. 

9.  What  is  the  square  root  of  0,0001234. 1 

Ans. 

10.  What  is  the  square  root  of  227  to  within  x^o o7  '^ 

*  Ans.  15,0665. 

11.  What  is  the  square  root  of  3,425  to  within  yiy  1 

Ans.  1,85. 

12.  What  is  the  square  root  of  fjj^f  1  Ansi  | J. 

13.  What  is  the  square  root  of  11}^  to  within  ,001 1 

Ans.  3,418. 

14.  What  is  the  Square  root  of  3  to  within  ,00000000011 

Ans.  1,7320508076. 

EXTRACTION   OF    THE    CUBE   ROOT   OF    NUMBERS. 

282.  The  third  power ^  or  cube  of  a  number,  is  the  product 
arising  from  multiplying  this  number  into  itself  till  it  has 
been  used  three  times  as  a  factor.  The  third  or  cube  root  is 
a  number  which,  being  raised  to  the  third  power,  will  pro* 
duce  the  proposed  number. 


164  ELEMENTS    OF  ALGEBRA.  [sECT.  VI. 

The  first  ten  numbers  being 

1,  2,  3,    4,     5,      6,      7,      8,      9,       10, 
Their  cubes  are,  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

The  numbers  of  the  first  line  are  the  cube  roots  of  the 
second. 

By  inspecting  these  lines,  we  perceive  there  are  but  nine 
perfect  cubes  among  numbers  expressed  by  one,  two,  or  three 
figures  ',  the  cube  root  of  other  numbers  consisting  of  one, 
two,  or  three  figures,  cannot  be  expressed  exactly  by  means 
of  unity,  as  may  be  shown  by  a  process  similar  to  that  used 
in  Art.  279. 

The  cube  root  of  an  entire  number  consisting  of  not  more 
than  three  figures,  may  be  obtained  by  merely  inspecting 
the  cubes  of  the  first  nine  numbers.  Thus,  the  cube  root  of 
125  is  5  ;  the  cube  root  of  30  is  3  plus  a  fraction,  or  within 
one  of  3. 

To  extract  the  cube  root  of  a  number  consisting  of  more 
than  three  figures,  we  present  the  following 

RULE. 

1.  Separate  the  given  number  into  periods  of  three  figures 
each,  beginning  at  the  right  hand :  the  left-hand  period  will  often 
contain  less  than  three  figures. 

2.  Find  the  greatest  cube  in  the  left-hand  period,  and  place  its 
root  on  the  right,  in  the  place  of  a  quotient  in  division.  Sub- 
tract the  cube  of  this  figure  of  the  root  from  the  first  period,  and 
to  the  remainder  bring  down  the  next  period,  afid  call  this  num- 
ber the  dividend. 

3.  Multiply  the  square  of  the  root  just  found  by  300  for  a 
divisor.  Find  how  many  times  the  divisor  is  contained  in  the 
dividend,  and  place  the  quotient  for  a  second  figure  of  the  root. 
Multiply  the  divisor  by  this  second  figure^  and  place  the  product 
under  the  dividend.  Multiply  the  former  figure  or  figures  of  the 
root  by  30,  and  that  product  by  the  square  of  the  last  figure, 
and  place  the  result  under  the  last ;  under  these  two  products 
place  the  cube  of  the  last  figure  of  the  root,  and  call  the  sum  of 
the  last  three  numbers  the  subtrahend. 


SECT.  VI.]  CUBE    ROOT    OF    NUMBERS.  165 

4.  Subtract  the  subtrahend  from  the  dividend^  and  to  the  re- 
mainder bring  down  the  next  period  for  a  new  dividend  ;  and  in 
finding  a  divisor  and  subtrahend^  proceed  precisely  as  before^ 
and  so  continue  till  all  the  periods  have  been  brought  down. 

We  proceed  next  to  the  explanation  of  this  tule. 

The  cube  of  a  binomial,  as  a-\-b=c^+2a'b^'^ab''-\-b\  In 
the  number  45,  a  may  represent  tens,  and  b  units,  or  we  max 
find  the  cube  of  45  writing  it  40+5. 

45=40-f-     5  . 
40-h     5  " 

200+  25 
1600+200 


(45^=1600+400+  25 
40+     5 

8000+2000+125 
64000+16000+1000 

(45)»=64000  +  24000  +  3000+ 125=91125 
283.  On  inspecting  the  above  examples,  it  will  be  per- 
ceived that  the  cube  of  a  number  composed  of  tens  and  units 
is  equal  to  the  cube  of  the  tens,  plus  three  times  the  prod- 
uct of  the  square  of  the  tens  by  the  units,  plus  three  times 
the  tens  by  the  square  of  the  units,  plus  the  cube  of  the 
units.  Let  it  now  be  required  to  reverse  the  above  process, 
and  find  the  cube  root  of  91125. 

(40^=64000,  (50)^=125000. 
Hence  the  cube  root  is  evidently  40  plus  a  certain  number 
of  units.  Subtracting  the  cube  of  40,  that  is,  the  cube  of  the 
tens,  there  remains  27125,  which  contains  the  remainder  of 
the  parts  above  specified.  As  it  is  evident  that  the  third 
power  of  tens  can  have  no  significant  figure  below  the  fourth 
place,  in  finding  the  third  root  of  the  tens  the  three  figures 
on  the  right  may  be  rejected,  as  they  will  not  influence  the 
result.  As  the  cube  of  100  is  1000000,  in  obtaining  the  cube 
root  of  hundreds  in  a  number  consisting  of  more  than  six 
figures,  we  may  reject  the  first  six  figures  on  the  right  as 


166  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 

not  influencing  the  result ;  hence  any  number  of  which  the 
root  is  required  may  be  separated  into  periods  of  three  fig- 
ures each,  each  one  of  the  periods  may  be  used  separately 
in  connexion  with  the  remainder  resulting  from  the  prece- 
ding operations,  and  the  number  of  periods  will  be  the  num- 
ber of  figures  in  the  root.  The  cube  of  no  one  of  the  digits 
.contains  more  than  three  figures.  In  the  above  example, 
rejecting  the  first  three  figures  on  the  right,  the  cube  root 
of  the  tens  found  in  91  is  4.  Subtracting  the  cube  of  this 
(64),  and  bringing  down  the  next  period,  the  result  of  the 
operation  is  27125.  This  must  contain,  from  what  has  been 
said,  triple  the  product  of  the  square  of  the  tens  by  the 
units,  together  with  two  remaining  parts  already  specified. 
As  the  square  of  tens  contains  no  significant  figures  less 
than  hundreds,  we  may  reject  the  two  right-hand  figures 
from  27125,  and  dividing  the  remainder  by  three  times  the 
square  of  the  tens,  Ave  should  obtain  the  unit  figure.  In 
practice  it  is  found  more  convenient  to  use  the  whole  divi- 
dend, and  to  annex  two  ciphers  to  the  divisor,  as,  instead  of 
multiplying  the  square  of  the  tens  by  three,  we  multiply  by 
300.  For  the  same  reason,  instead  of  multiplying  the  prod- 
uct of  the  square  of  the  units  by  the  tens  by  3,  we  multiply 
by  30. 

Dividing  27125  by  the  square  of  the  tens  (16),  multiplied 
by  300,  which  =4800,  the  quotient  5  will  be  the  unit  figur.e 
sought ;  or  it  may  be  too  large  by  1  or  2,  as  there  may  be 
hundreds  arising  from  the  other  parts  of  the  root  sought : 
this  can  only  be  determined  by  trial.  Having  now  the  tens 
and  the  units,  and  having  already  subtracted  the  cube  of  the 
tens,  we  next  proceed  to  subtract  the  other  parts  of  the 
cube  from  the  remainder. 

The  square  of  the  tens,  multiplied  by  300  and  by  the  units, 
the  last  figure  of  the  root       -         -         -         -    =24000 

The  tens,  multiplied  by  the  square  of  the  units 

and  by  30 =3000 

The  cube  of  the  last  figure  or  units    -         -         -      =   125 

Sum        -        -       27125 


SECT.  VI.]        CUBE  ROOT  OF  NUMBERS.     •       167 

As  there  is  no  remainder,  45  is  the  root. 
The  operation  may  be  exhibited  as  follows : 

91125:45 

(4)'=  64 '~~ 

(4)*=  16)  X  300=4800)27125 
4800  X     5=  24000 

(5)«x4x    30=  3000 

(5)'=5x5x5=  125 

000 
284.  Any  number,  however  large,  may  be  considered  as 
composed  of  units  and  tens  :  the  process  of  finding  the  cube 
root  may  therefore  be  reduced  to  that  of  the  preceding  ex- 
ample. 

Required  the  third  root  of  9663597. 

9663597!213,  root. 
(2)»=  8  ' 

Di » imr. T" 

(2)*x  300=  1200)  166^,  first  dividend. 
1200x1=  1200 

2x30x(l)'=  60 

(1)'=         L 

^_^    1261,  first  subtrahend. 
(21^x300  =  132300)402597,  second  dividend. 
132300x3=  396900 

21x30x(3^)=  5670 

(3^)=  27^ 

402597,  second  subtrahend. 
OOOOW 
Should  the  divisor  not  be  contained  in  the  dividend  as 
prepared  above,  place  a  cipher  in  the  root,  and  bring  down 
the  next  period  to  form  a  new  dividend. 

The  difference  between  the  cubes  of  two  consecutive  whole  num- 
hers  is  equal  to  three  times  the  square  of  the  least  number^  plus 
three  times  this  number ^  plus  1. 

Let  a  and  a-j-1  be  two  consecutive  whole  numbers. 
(a+l)*=tf'+3o'+3a-|-l. 
(a-f-1)'— a'=3a'  +  3a+l. 
(90)»-(89)*=3  X  (89)^+3  X  89+ 1  =24031. 
In  extracting  the  cube  root  of  any  number  not  a  perfect 


168  ELEMENTS    QF    ALGEBRA.  [sECT.   YI. 

cube,  if  any  of  the  remainders  are  equal  to,  or  exceed  three 
times  the  square  of  the  root  obtained,  plus  three  times  this 
root,  plus  1,  the  last  figure  of  the  root  is  too  small,  and  must 
be  augmented  by  at  least  unity. 

285.  The  third  root  of  a  fraction  is  found  by  extracting 
the  third  root  of  the  numerator  and  denominator.  When 
the  denominator  is  not  a  perfect  third  power,  we  may  ob- 
tain the  root  approximately  by  multiplying  both  terms  by 
the  square  of  the  denominator  j  thus,  in  obtaining  the  cube 
root  of  ^,  we  multiply  both  terms  by  49  J  the  fraction  then 
becomes  i|^|,  the  root  of  which  is  nearest  ^  accurate  to 
within  ^.  We  might  multiply  both  terms  of  ^^^  by  any 
perfect  cube,  and  then  extract  the  cube  root,  and  we  should 
approximate  still  nearer  the  true  rooti  By  a  process  similar 
to  that  explained  in  the  article  on  square  root,  we  may  ap- 
proximate the  third  root  of  a  number  not  a  perfect  third 
power,  by  converting  it  into  a  fraction,  the  denominator  of 
which  is  a  perfect  third  power.  Thus  the  approximate  root 
of  15  may  be  found,  putting  it  under  the  following  form ; 

15x12=^=25920 
(12f        1728  ' 
the  third  root  of  which  is  f  |,  or  2^2  accurate  to  within  less 
than  j^j.     The  root  may  be  obtained  with  greater  accuracy 
by  using  some  number  greater  than  12. 

In  such  cases  it  is  most  convenient  to  convert  the  propo- 
sed number  into  a  fraction,  the  denominator  of  which  shall 
be  the  third  power  of  10,  100,  1000,  &c.  Let  it  be  required 
to  find  the  third  root  of  25  to  within  ,001  ;  converting  25 
into  a  decimal,  the  denominator  of  which  is  the  third  power 
of  1000,  viz.,  25,000  000000,  the  third  root  of  which  is  2,920 
to  within  ,001,  we  have  then  v^  25=2,920  accurate  to  with- 
in less  than  ,001. 

To  approximate  the  third-  root  of  an  entire  number  by 
means  of  decimals,  we  annex  to  the  proposed  number  three  times 
as  many  ciphers  as  there  are  decimal  places  required  in  the  root ; 
we  then  extract  the  root  of  the  number  thus  prepared  to  within  a 


SECT.  VI.]        ROOTS  OF  ANY  DEGREE.  160 

timV,  and  point  off  for  decimals  as  many  places  as  there  are  deci- 
tnal  figures  required  i?i  the  root. 

If  the  proposed  number  contain  decimals,  beginning  at  the 
place  of  units,  separate  the  number,  both  to  the  right  and 
left,  into  periods  of  three  figures,  annexing  ciphers,  if  ne- 
cessary, to  complete  the  right-hand  period  in  the  decimal 
part.  Then  extract  the  root,  and  point  off  for  decimals  in 
the  root  as  many  places  as  there  are  periods  in  the  decimal 
part  of  the  power. 

The  third  root  of  a  vulgar  fraction  may  be  most  readily 
obtained  after  converting  it  first  into  a  decimal  fraction. 

EXAMPLES. 

1.  What  is  the  cube  root  of  75686967  1  ^ns.  423. 

2.  What  is  the  cube  root  of  128787625  1         ^ns.  505. 

3.  What  is  the  cube  root  of  2054.83447701 1  ^ns.  5901. 

4.  What  is  the  cube  root  of  52458 1674,6251  Ans,  806,5. 

5.  What  is  the  cube  root  of  1003,003001 1     Ans.  10,01. 

6.  What  is  the  cube  root  of  0,756058031  \    Ans,  0,911. 

7.  What  is  the  cube  root  of  32977340218432  % 

Ans,  32068. 

8.  What  is  the  cube  root  of  473  to  within  ^V  '^-     •^^*-  ^h 

9.  What  is  the  cube  root  of  79  to  within  ,0001 1 

Ans.  4,2908. 

10.  What  is  the  cube  root  of  3,00415  to  within  ,0001  \ 

Ans.  1,4429. 

11.  What  is  the  cube  root  of  0,00101  to  within  ,01  \ 

Ans.  0,10. 

12.  What  is  the  cube  root  of  0,000003442951 1 

Ans.  0.0151. 

13.  What  is  the  cube  root  of  6iff f  1  Ans.  Iff 

14.  What  is  the  cube  root  of  iJfH  '^  *^^^'  f  f • 

TO  EXTRACT  ANY  GIVEN  ROOT  OF  A  WHOLE  NUMBER. 

286.  Any  root  exceeding  the  third,  consisting  simply  of 

two  and  three,  as  factors,  may  be  found  by  the  preceding 

rules  j  thus,  the  fourth  root  may  be  found  by  extracting  the 

square  root  twice ;  the  sixth  root  by  extracting  the  third 

15  Y 


170  ELEMENTS  OF  ALGEBRA.  [sECT.  VI. 

root,  and  then  the  square  root  of  that ;  the  twelfth  root  hy 
extracting  the  square  root  twice,  and  then  th^  third  of  the 
last  root.  Before  proceeding  to  give  a  rule  for  the  extrac- 
tion of  any  root,  we  subjoin  a  table  of  roots  and  powers. 


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SECT.  VI.]        ROOTS  OF  ANY  DEGREE.  171 

{a-\-bY=a'-\-ba*b-{-  10a'ft'+  10a'lr'-^Dab*-^b\ 
{a-{-by=d'+la'b-\-2la'b'-{-S5a*lf'-^3ba'b*-\-2la'b'-^lab'+b\ 

In  these  examples  a  and  b  may  represent  tens  and  units 
in  any  given  number,  as  47  j  and  to  obtain  the  root  of  any 
given  power,  we  evidently  must  reverse  the  process  by  which 
the  power  is  obtained  from  the  root. 

By  carefully  attending  to  the  preceding  explanations,  and 
the  different  powers  of  the  binomial  (a-{-b),  the  reason  of  the 
following  rule  for  extracting  any  given  root  of  a  proposed 
number  will  readily  be  discovered: 

1.  Divide  the  number  into  periods  of  as  many  figures  each  as 
there  are  units  in  the  index  denoting  the  root, 

2.  Find  the  first  figure  of  the  root  by  trials  and  subtract  its 
power  from  the  left-hand  period^  and  to  the  remainder  bring  down 
the  first  figure  of  the  next  period  for  a  dividend. 

3.  Involve  the  root  to  the  next  inferior  power  to  that  which  is 
given,  and  multiply  it  by  the  number  denotijig  the  given  power^ 
and  it  will  be  the  divisor. 

4.  Find  how  many  times  the  divisor  is  contained  in  the  divi- 
dend, and  the  quotient  will  be  another  figure  of  the  root,  or  1  or 
2  too  large. 

5.  Involve  the  whole  root  to  the  given  power,  and  subtract  it 
from  the  two  left-hand  periods  of  the  given  number;  bring  down 
the  first  figure  of  the  next  period  to  the  remainder  for  a  new  div' 
idend,  find  a  new  divisor,  another  figure  of  the  root,  and  again 
involving  the  whole  root  to  the  given  power,  subtract  it  from  the 
first  three  left-hand  periods.  Thus  proceed  till  the  whole  root  is 
obtained.  • 

Required  the  fifth  root  of  36936242722357. 

36936242722357|517 
5^=  3125 

5*  X  5  =  3125,  first  divisor.  5686,  first  dividend. 

(51)*=  345025251,  subtrahend. 

(51)*  X  5  =  33826005,  2d  divisor.        243371762,  2d  dividend, 
(517)*=  36936242722357 

0000000 


172  ELEMENTS    OF    ALGEBRA.  [sECT.  VI. 

287.  The  preceding  rule  may  be  put  in  another  form,  em- 
bracing the  same  principles,  but  more  consistent  with  the 
method  by  which  we  have  explained  the  extraction  of  the 
second  and  third  roots. 

RULE. 

1.  Separate  the  numler  into  periods  of  as  many  figures  as 
there  are  units  in  the  index  denoting  the  rooti 

2.  Find  by  trial  the  root  of  the  first  period :  this  will  he  the 
first  figure  of  the  root :  place  this  figure  to  the  left^  in  a  column 
called  FIRST  column  ;  then  multiply  it  by  itself  and  place  the 
product  for  the  first  term  of  a  second  column.  This,  multi- 
plied by  the  same  figure,  will  give  the  first  term  of  a  third  col- 
umn. Thus  continue  until  the  number  of  columns  is  one  less 
than  the  units  in  the  index  denoting  the  root.         , 

Multiply  the  term  in  the  last  column  by  the  same  figure,  and 
subtract  the  product  from  the  first  period,  and  to  the  remainder 
bring  down  the  next  period,  and  it  will  form  the  first  dividend. 

Jlgain,  add  this  same  figure  to  the  term  of  the  first  column, 
multiply  the  sum  by  the  same  figure,  and  add  the  product  to  the 
term  of  the  second  column,  which,  in  turn,  must  be  multiplied- 
by  the  same  figure,  and  added  to  the  term  of  the  third  column,. 
and  so  on  till  we  reach  the  last  column,  the  term  of  which  will 
form  the  first  trial  divisor. 

Jlgain,  beginning  with  the  first  column,  repeat  the  above  pro- 
cess until  you  reach  the  column  next  to  the  last  ;  and  so  continue 
to  do  until  there  are  as  many  terms  in  the  first  column  as  there 
are  units  in  the  index  denoting  the  root,  observing  in  each  suc- 
cessive operation  to  terminate  in  the  column  of  the  next  inferior 
order. 

3.  Seek  how  many  times  the  first  trial  divisor,  when  there  are 
annexed  to  it  as  many  ciphers,  less  one,  as  there  are  units  in  the 
index,  is  contained  in  the  first  dividend  ;  the  quotient  figure 
will  be  the  second  figure  of  the  root. 

Then  proceed  to  form  a  new  series  by  annexing  this  figure  to 
the  last  term  in  the  first  column  ;  multiply  the  result  by  the  last 
figure,  and  add  it  to  the  last  term  in  the  second  column,  advan- 


SECT.  VI.]         ROOTS  OF  ANY  DEGREE.  173 

cing  the  number  to  be  added  two  places  to  the  right  of  the  other 
before  adding.  Multiply  this  result  by  the  same  figure^  and  add 
the  product  to  the  last  term  in  the  third  column,  having  previ- 
ously advanced  it  three  places  to  the  right  of  that  term  ;  proceed 
in  the  same  manner  to  the  last  term^  observing  to  advance  the 
numbers  added  to  the  different  columns  as  many  places  to  the 
right  of  the  terms  as  the  number  expressing  the  order  of  the  col- 
umn  ;  that  iSy  advancing  the  terms  of  the  first  column  one  place, 
those  of  the  tEcpND  column  two  places,  Src  Multiply  the  term 
thus  obtained  in  the  last  column  by  the  last' figure  of  the  root, 
and  subtract  it  from  the  dividend  ;  to  the  remainder  bring  down 
the  next  period  for  a  new  dividend,  and  proceed  to  find  a  divisor 
and  the  third  figure  of  the  root  in  the  same  manner  as  the  second 
was  obtained. 

Proceed  in  the  same  manner^till  all  the  periods  are  brought 
down.  If  there  is  still  a  remainder,  the  process  can  be  extended 
by  forming  periods  of  ciphers. 

Required  the  third  root  of  103823. 


Ist  col. 
4 
8 

127 

1 

2</  col. 
16 

48,  first  trial  divisor. 
5689 

10382347 
64       '- 
39823,  first  div. 
39823 

288.  It  will  be  perceived  that  this  method  of  extracting 
the  cube  root  is  similar  to  that  already  explained.  Let  a-}- 
J=40-h7=47(a+Z»f =a»+3a'A+3aZ»»  +  ^^  We  subtract  first 
the  cube  of  the  tens,  a*=64.  We  next  form  the  divisor, 
which  is  3a'=48  ;  16,  the  first  term  of  the  second  column, 
is  once  the  square  of  the  tens  ;  8,  the  second  term  of  the 
first  column,  is  double  the  tens  j  multiplying  this  by  the 
tens  (4),  the  product  is  twice  the  square  of  the  tens  (32) ; 
this  added  to  16,  the  square  of  the  tens,  gives  48,  three  times 
the  square  of  the  tens.  Instead  of  rejecting  two  figures  on 
the  right  of  the  dividend,  we  annex  two  ciphers  to  the  divi- 
sor. The  second  figure  of  the  root  is  the  result  of  the  di- 
vision.    Then  there  remains  to  be  obtained  and  subtracted 


174 


ELEMENTS    OF    ALGEBRA. 


[sect.  vr. 


from  the  given  number^  3a^b-\-3ah'^-\-h^ :  this  may  be  put  in 
another  form,  thus  :  ( {3a-\-b)  x  b-\-3a^)  x  b.  On  inspecting  the 
above  work,  it  will  be  perceived  that  127=3a+&;  this,  mul- 
tiplied by  7  (b),  becomes  S89  =  {3a-\-b)  X  b  ;•  adding  4800 ^Sa^, 
the  result  is  5689  ;  multiplying,  according  to  the  rule,  by  7 
(b),  we  have  39823=( (3a+i) x^>+3a') x^;.  The  same  expla- 
nation will  apply,  however  extended  the  operations  may  be. 
Required  the  fifth  root  of  36936242722357. 


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What  is  the  seventh  root  of  1231171548132409344 '? 


SECT  TI.] 


ROOTS  OF  ANY  DEGREE. 


175 


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176  ELEMENTS    OF    ALGEBRA.  [sECT.  VI. 

EXAMPLES.' 

1.  Find  the  fifth  root  of  418227202051.  Jins.  211. 

2.  Find  the  fourth  root  of  75450765,3376.       Ans.  93,2. 

3.  Find  the  fifth  root  of  0,000016850581551. 

^7X5..  0,1 11. 

4.  Find  the  fourth  root  of  25^6,88187761.       Ans.  7,09. 

5.  Find  the  sixth  root  of  2985984.  Ans.  12. 

6.  Find  the  eighth  root  of  1679616.  Ans.  6. 

7.  Find  the  seventh  root  of  2.         Ans.  1,10409,  nearly. 

EVOLUTION  OF  MONOMIALS, 
* 
289.  From  a  previous  demonstration  (Art.  156),  it  is  evi- 
dent that  the  root  of  the  product  of  two  or  more  factors  is  equal 
to  the  product  of  the  roots.     Thus,  \/  a^b*c^z=  Va^xVtf^xV  c\ 
Again,  by  the  definition  of  evolution  (Art.  274),  \/c^=c^', 
for  c'xc^'^c^^^c';  hence   v/"?=c^-^^  or  ci=c\ 

And,  ^~Sc'=2c^ ',  for  2c''x'2c'x2c'=Sc'  -,  hence  ^Sc'zzz 
^8"x  ^7'=2xc''r-\  or  2(^=^c\. 

The  same  reasoning  will  evidently  apply  to  every  case  of 
monomials.    Hence,  for  the  evolution  of  monomials,  we  have 
^the. following  general 

J?:.  RULE.  t. 

1.  Extract  the  required  root  of  the  coefficient. 

2.  Divide  the  exponent  of  each  literal  factor  by  the  number 
denoting  the  root,  and  annex  the  result  to  the  root  of  the  coeffi- 
cient. 

JVote  1. — "With  regard  to  the  sign  to  be  prefixed  to  the 
root,  it  is  important  to  observe,  *  ^ 

a.  An  odd  root  of  a  number  will  have  the  same  sign  as  the 
number  itself.  Thus,  the  cube  root  of  a^,  or  ^a^=^a,  for  ax 
axa—a^\  and  the  cube  root  of — a^,  ox  ^  —a  = — a,  for — ax 
— flX  —a=—a^, 

b.  The  even  root  of  an  affirmative  number  is  ambiguous. 
Thus,  the  square  root  of  a'',  or  ^fa=±a;  {or  a  Xa=a\  and 
, — ax — a=+a  ;  also,  the  square  root  of  16,  or  \/16=±4, 
for  4x4=16,  and  -4x-4=  +  l6. 


SECT.  VJ.]       EVOLUTION  OF  MONOMIALS.  1T7 

c.  The  even  root  of  a  negative  number  is  impossible.  Thus, 
the  square  root  of  — a',  or  V — a*,  can  be  neither  +«  nor 
— a,  for  -\-aX'{-a=-^a^f  and  — aX — a=z-\-a^.  Also,  the 
square  root  of — 16,  or  V — 16,  can  be  neither  +4  nor  — 4. 

J^ote  2.— The  root  of  a  fraction  is  equal  to  the  root  of  the 
numerator  divided  by  the  root  of  the  denominator.     Thus, 


v\ 


-=— :  for  — X  — = 3C-. 

c     ck  c*     ci     c^i     c 


^ote  3. — The  above  rule  for  the  evolution  of  monomials 
is  equally  applicable  where  the  exponents  are  negative. 
Thus,  the  square  root  of  a~^,  or  y/ a~*=ar^  =a~^ ;  for  a~*=: 


I;  hence,  ^a-^=^l=l=a-\ 
a*  V   a*     or 


EXAMPLES. 

1.  Bequired  the  square  root  of  da*b^, 

2.  Required  the  square  root  of  64a*a?*.  Arts.  Sa'oc^, 

3.  Required  the  cube  root  of  21a'b\  A?is.  3a'b. 

4.  Required  the  fourth  root  of  16a^a?'y.  Ans.  2aVy*« 

5.  Required  the  square  root  of  .- —  Aits.  -^. 

^                   ^                      9a?y  3a:/ 

6.  Required  the  fifth  root  of  243a'°6^  Ans.  Sa'b. 

7.  Required  the  fourth  root  of  16o-^Z>*.  Ans.  2a-^b. 

8.  Required  the  sixth  root  of  64a®a:'y.  Ans.  2ax^yh, 

9.  Required  the  third  root  of  Sa-^b-^c^,  Ans.  2ar'b-^c, 

10.  Required  the  square  root  of  196a*6V.  Ans.  14a^i~'c*. 

11.  Required  the  square  root  of  784x^2*'.  Ans.^Sxi^z^. 

12.  Required  the  square  root  of  — J-.    '  Ans.  — ^. 

^                  ^                     496V  7k» 

13.  Required  the  cube  root  of  —Tta^b^.  Ans.  ~3a^b^. 

14.  Required  the  nih.  root  of  a"J*'c~*".  Ans.  aV^cr*. 

15.  Required  the  fifth  root  of  — 32a'^6'V*.  Ans.  —1ah^(?. 

16.  Required  the  fourth  root  of  Sla^^^V.  Ans.  3(iShc^ . 

17.  Required  the  cube  root  of  — 64a~^Z»"^c-'^ 

Ans.  — 4a-'J-*(r^. 
Z 


178  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 

18.  Required  the  square  root  of  44 la?^3/V.     ^ns.^lx'^yz^, 

19.  Required  the  square  root  of  576a^^>-^c-''c?'^ 

20.  Required  the  fifth  root  of  — 243a?-'2/'°;s-»^ 

Ans.  —3x-yz-\ 

21.  Required  the  square  root  of .  Ans.  — ^^. 

290.  If  all  the  factors  of  which  the  monomial  is  composed 
are  not  complete  powers  of  the  same  name  as  the  root,  it  is 
evident  that  the  root  of  the  entire  number  cannot  be  obtain- 
ed. Still  the  expression  may  he  simplified  by  removing  that 
factor  which  is  a  complete  power  of.  the  same  name  as  the 
required  root,  from  under  the  radical  sign.  This  is  done  on 
the  principle  that  the  root  of  the  product  is  equal  to  the  product 
of  the  roots. 

Thus,  V 8^  =  v/4a2x  26==  v/4?x  \/'2i  =  2a  V26. 


And,    ^2Ub'=4^Sb'x3a='^Sb^x^3a=2b  VSa. 


And,    V6a"Z>  =ya"  x^b=Va''   xVQb=:a    V6b. 
Hence,  to  reduce  radicals  to  their  simplest  forms : 

1.  Resolve  the  quantity  under  the  radical  sign  into  two  faC' 
tors,  one  of  which  shall  be  a  complete  power  of  the  same  name  as 
the  root. 

2.  Extract  the  root  of  this  factor,  and  multiply  it  by  the  co- 
efficient of  the  radical f  if  it  has  any,  and  prefix  the  result  to  the 
radical  sign  under  which  the  factor  that  is  not  a  complete  power 
will  remain. 

EXAMPLES. 


Required  the  simplest  form  of  \/S2a'*b'^c. 

Ans.  U^by/Yc, 


2.  Required  the  simplest  form  of  ^/^Sa^¥c'^d, 

Ans.  '7ab^cW'2d, 


3.  Required  the  simplest  form  of  v^24a^c^^ 

Ans.  2ad'VSc, 


^.  Required  the  simplest  form  of  \/ b4^a^xy'-^z\ 

*  Ans.  3aSjz^V~^. 


SECT.  VI.]  EVOLUTION    OF    MONOMIALS.  179 


5.  Required  the  simplest  form  of  VS2a*b'c. 

Jlns.  QaVV^c, 

6.  Required  the  simplest  form  of  a/— ^. 

V    4fox  y 

V  48x^y     V  16x*     3y     V  16x*     V  3y     ^x'     V  3y 

2^  V    3^* 


Jlns.  ^^f. 


7.  Required  the  simplest  form  of  V^a^b—l^a^x. 


Jlns.  2aN/2A— 3x. 


v/8a»6- 12a«x=  v/4a2 X  (2A-3x)=  v/4o^ X  x/26-3a:=2a 
V2^3^ 


8.  Required  the  simplest  form  of  ^^/.JlfL— 


Ans.  ^l/E 
3c  \^    d 


9.  Required  the  simplest  forip  of  ^24a'c— 32tt'cx. 


Ans.  2a^3c— 4cx. 
10.  Required  the  simplest  form  of  y/c^-^a^b^. 

Ans.  a^l+^. 


11.  Required  the  simplest  form  of  \/^Oba^b*c^de. 
v'405a»6V(/e=  v/81a'6Vx  v/5^=9a*»Cv/"Wer 


^/w.  9a^CN/5a(ie. 


12.  Required  the  simplest  form  of  '^QOba'b^c'd^. 


Ans.  lWb^(^ds/bacd. 


13.  Required  the  simplest  form  of  v^lOUaVc'd 


Ans.  13a«6*cv/6aicd 


14.  Required  the  simplest  form  of  V —Sd'x. 
v/-8a«x=v/4a'x-2j;=v/4?Xv/"^^^=2av/^r2i. 


Ans.  2av/— 2a:. 


15.  Required  the  simplest  form  of  v/  — 16. 

Ans.  4v/^n". 


180  *       ELEMENTS    OF  ALGEBRA.  [sECT.  VI. 

EVOLUTION  OF  POLYNOMIALS. 
291.  We  might  give  rules  for  the  extraction  of  the  differ- 
ent roots  separately,  but  it  will  comport  better  with  our 
purpose  to  introduce  the  student  at  once  to  a  general  rule 
by  which  we  may  evolve  any  root  whatever.  The  reason 
for  the  following  rule  will  be  sufficiently  obvious  if  we  re- 
cur to  the  formation  of  powers  by  the  binomial  theorem  or 
by  actual  multiplication  ;  and,  indeed,  the  work  verifiear  itself. 

•     RULE. 

1.  Arrange  the  terms  according  to  the  powers  of  one  of  the 
letters^  so  that  the  highest  power  shall  stand  in  the  first  term,  the 
next  highest  in  the  second,  Sfc. 

2.  Find  the  root  of  the  first  term,  and  place  it  in  the  quotient ; 
then  subtract  its  power  from  the  first  term,  and  Iring  down  the 
second  term  for  a  dividend. 

3.  Involve  the  first  term  of  the  root  to  the  next  inferior  power, 
and  multiply  it  by  the  index  of  the  given  power  for  a  divisor. 
Divide  the  dividend  hy  this  divisor,  and  the  quotient  will  he  the 
second  term  of  the  root. 

4-.  Involve  the  terms  of  the  root  thus  found  to  the  given  power, 
and  subtract  it  from  the  whole  polynomial.  Divide  the  first  term 
of  the  remainder  by  the  divisor  first  found  ;  the  quotient  will  be 
another  term  of  the  root. 

5.  Proceed  in  this  manner  till  the  power  obtained  by  the  invo- 
lution of  the  terms  of  the  root  is  equal  to  the  given  polynomial. 
This  will  be  the  case  only  when  the  true  root  is  found. 

EXAMPLES. 

1.  Kequired  the  square  root  of  4a^-{-4ai-f  6^ 
4>a^-\-^ab-{-b\     \2a-{-b.     Ans. 
4.a^ 


4a)  *  -\-^<ab^ 
4a=^+4a64-6' 


0 
2.  Required  the  cube  root  of  a -|-3a''— 3a^  — lla'+6a^-j- 
12a-8. 


8BCT.  VI.]  EVOLUTION   OF   POLYNOMIALS.  181 

oe^3a5«3a*-lla*+6a»+12a-8     |a«+a-2.    Jlns. 


3a*)*  +3a» 


a«+3a*+3a*H-a* 
3a0*       •  -6a*— 12a=' 


a6_|-3a6_3a*-lla^+6a'+l:2a-8. 


0 
3.    Required  the    fourth   root  of   16a*+96a»6+216a«6«-f- 
216air»-f81J*. 
16a*4-96a'64-216a*6*+216ai'H-81J*.     |2a+3^.     Ans. 

16a* 


32a')     •  +96a»J 


16a*+96a'i4-216a«6»4-216a^>'+8l6*. 

'  0 

4.  Required  the  cube  root  of  a«— 6a^64-15a*J*— 20a'^>'+ 
15a'6*— 6a5»H-6^  Ans,  a^—'lab  +  h''. 

5.  Required  the  fifth  root  of  32a»-80a*a:H-80aV-40aV 
+  lOox* — X*.  Ans,  2a — x, 

6.  Required  the  fifth  root  of  a'  +  5a*6+10a'6»4-10a*J'+ 
5aZ>*-f  J*.  ^  Ans,  a +  6. 

7.  Required  the  sixth  root  of  a»— 6a«6+15a*^— 20a='6»+ 
l^a^b^—%ab^-\-b\  Ans,  a—b. 

292.  Remark  1. — The  square  of  a  binomial  consists  of 
three  parts,  viz.,  the  square  of  the  first  term,  twice  the  prod- 
uct of  the  two  terms,  and  the  square  of  the  last  term.  Hence 
the  second  power  of  the  simplest  polynomial  will  consist  of 
three  terms ;  and  every  trinomial  in  which,  when  the  terms 
are  arranged,  the  extremes  are  complete  squares,  and  the 
middle  term  is  double  the  product  of  the  square  roots  of  the 
extremes,  is  a  perfect  square,  whose  root  may  be  found  by 
the  following 

BI7LE. 

Take  the  square  roots  of  the  two  terms  that  are  complete  pow^ 
erSj  and  connect  them  by  the  sign  prefixed  to  the  other  term* 
16 


182  ELEMENTS    OF    ALGEB*RA.  [sECT.  VI. 


1.  sfa"-    -i-2ab    i-b^    =a  +h. 


2.  Va"    —^ah    +6^    =a  —5. 


3.   x/a^    +2a      +1     =0+1. 


V  3  9  3 


5.  x/36y2+36y  +9     =6^+3. 

6.  V^d^  -Qdk   +A2   =:3c?-A. 


7.  v/a'Z>2+2ak(^+c2t/2-a6+cJ. 

8.  ./^-^    +-^  ..^  -1 

293.  JRemark  2. — Since  the  fourth  power  of  a  quantity- 
may  be  found  by  squaring  the  second  power,  it  is  evident 
that  the  fourth  root  may  be  obtained  by  extracting  the 
square  root  of  the  square  root. 

Thus,  V^*=\/ V^=y/^=a. 

And,    V^'=\/v~^=V^^=a,Scc. 
Hence, 

1.  To  obtain  the  fourth  rootj  we  may  extract  the  square  root 
of  the  square  root. 

2.  To  obtain  the  sixth  root,  we  may  extract  the  cube  root  of  the 
square  root,  Src.  • 

>  EXAMPLES. 

1.  Required  the  square  root  and  fourth  root  of  16a'' -j- 
96(^b+'21QaW-\-216ab^+Slb\ 

Ans,  4a'+12a6+95^  and  2a+3J. 

2.  Required   the  sixth  root   of  a;'— 12a?^+60a?''— 160ar'+ 
240a:'— 192a;+64.  *^ns.  a?— 2. 

3.  Required  the  eighth  root  of  a^-\-^a'b+'^Sa%''+bQa'b^-\' 
10a''¥+bQa^b'-{-2'^a%^+^aV^b\  Ans.  a+b, 

4.  Required  the  ninth  root  of  a^+9a«i+36a'Z»2+84a^5''-{- 
126a^6^+126a^55+ 84^=^6^+ 36a2Z>'-h9a5« 4- *'.  Ans.  a^b. 

294.  Remark  3. — If  the  polynomial  is  not  a  perfect  pow- 
er, it  may  sometimes  be  simplified  in  the  same  manner  as 
monomials.  ^ 


SECT.  VI.]  EVOLUTION   OF   POLYNOMIALS.  183 


1.  Required  the  simplest  form  of  y/5(^-\-l0ab-\-5tl^. 


2.  Required  the  simplest  form  of  \/o'6+4a'^-f  4o^' 

jJns.  {a-{-2b)y/ab. 


3.  Required  the  simplest  form  of  V2a'+8a*ft+12a*6*4- 

4.  Required  the  simplest  form  of  VSa*b—9a^t^-{-9a^l/^ — 
3a^.  •^ns.  (a—b)V3ab. 

295.  Remark  4. — Roots  may  also  be  obtained  by  the  Bino- 
mial Theorem,  since  n  in  the  general  formula  (Art.  268)  may 
be  either  an  integer  or  a  fraction. '  The  series  produced  by 
the  expansion  of  a  binomial,  however,  will  never  terminate, 
since  the  successive  subtractions  of  unit's  from  the  fraction- 
al exponent  of  the  leading  letter  can  never  reduce  that  ex- 
ponent to  0. 

EXAMPLES. 

1.  Expand  by  the  Binomial  Theorem  (a-\-by. 
The  exponents  in  the  successive  terms  of  the  result  will 
be  as  follows : 
Of  a 

Of  J 

0,  1,  2,  3,  4,  &c. 

Hence  the  letters,  without  their  coefficients,  will  be 

J^arh+a~h--{-a~^l/'-^a~h\  &c.,  ad  infin. 
Proceeding  as  in  Art.  271,  we  shall  obtain  for  the  coeffi- 
cients of  the  successive  terms, 
*»  3>  2  X  — ^"r2= — £4,  — 2^X  — f  "^"=2X6,  iisX     5^T-4«=— 

8,4^8'  ****" 

Or         -         1,  i,  —I,  tV  — ihy  &^c-»  «^  ^^fi^' 
Hence,  by  compounding  the  series  of  letters  and  coeffi- 
cients, we  obtain 

(a+»)*=ai+d*_i2*!+fd^_?±',  &c,  ad  infin. 
^        '  8  8  16        128 '        '  •' 


184  ELEMENTS  OF  ALGEBRA.  [sECT.  VI. 

Transferring  a,  where  it  is  affected  with  the  negative  ex- 
ponent, to  the  denominator,  as  in  Art.  258,  the  expression 
becomes 

x  1  b  b''  V  b' 

Ca-\-by=a^-{- — 1 — — 3-1- 5 — Y,  &c.,  adinjin, 

2.  Expand  (a—b)^.  *, 

The  exponents,  obtained  as  in  the  last  example,  are  as 
follows : 

Of/7  1     1 1— 3      ^ 1— 5.    S 1— 8     ^f. 

Of  6 0,  1,        ^        2,  3,  &c. 

Hence  the  letters,  &c.,  will  be 

ai—a~^b-\-a-^b^—am\  &c.  adinjin. 
The  coefficients,  obtained  as  before,  are 

1,  h  ix— 1-^2=— 3-e,  — 34x— f--3=3^,  &0. 
Or      -        -       l+^—re+Hgj  &CC.,  ad  infin. 

1      ±    A-        ^^'         2.5Z>" 
Hence  -   ^a-b)^=a^^^^-^^^^j-^^^^^j,6cc.,  adinjin. 

EXAMPLES. 

1.  Expand  by  the  Binomial  Theorem  (a-^b)^. 

Diminishing  the  exponents  ^  successively  by  1,  the  expo- 
nents of  a,  in  the  successive  terms,  are  i,  — ^,  — |, — |,  &c. 
(Art.  271.) 

The  coefficients  obtained  by  the  general  theorem  (Art. 
271)  are  as  follows :  1,  i,  i X -_|-7-2=:— |,  — } x— f -^3=yV• 


2.  Expand  (a-^bf. 

3.  Expand  ( I +a:)^. 

4.  Expand  (a+&)^. 

5.  Expand  (a — b)^. 


6.  Expand  (a— a?)"^. 

7.  Expand  b{a^—b)-\ 

8.  Expand  (a^-bx)-^. 

9.  Expand  V2=(l  +  1)^ 


CALCULUS  OF  RADICALS. 

296.  A  radical  quantity  is  the  indicated  root  of  an  imper- 
fect power;  as,  \/2,  \/a,  and  \/b. 

Radicals  are  similar  when  they  are  composed  of  the  same 
numbers  or  letters,  placed  under  the  same  radical  sign  or 
index.     Thus,  v/a,  4v/o,  and  a^a  are  similar  radicals* 


SBCT.  TI.]  CALCULUS   OF    RADICALS.  185 

Before  entering  upon  equations  of  the  higher  degrees,  we 
will  consider  some  of  the  transformations  that  may  be  made 
upon  algebraic  expressions  involving  radicals. 

CASE   I. 

297:  To  reduce  a  rational  number  to  the  form  of  a  radioed, 

RULE. 

1.  Involve  the  given  number  to  a  power  of  the  same  name  as 
the  root. 

2.  *^pply  the  corresponding  radical  sign  or  index  to  the  power 
thw  produced, 

1.  Reduce  3a  to  the  form  of  the  fourth  root. 

3a  involved  to  the  fourth  power  equals  81o*. 
Applying  the  radical  sign,  3a=V81a*j  or  applying  the 

fractional  index,  3a=(81a*)*^. 

2.  Reduce  5c^b  to  the  form  of  the  third  root. 


Jins.  Vl25a'^,  or  {I25a«i»)^. 

3.  Reduce  2ax'y*  to  the  form  of  the  eighth  root. 

Jlns.  4^2560^3^,  or  (256a«x*V"A 

4.  Reduce  {a^cx^  to  the  form  of  the  fourth  root. 

^ns,  ^^ya'cV",  or  {-^ja'c'x'^)K 

6.  Reduce  _— _  to  the  form  of  the  third  root. 
Wd-'if 

Ans    V-?Z^!^'  or  /27aW\.} 

CASE  II. 

298.  To  introduce  a  rational  coefficient  under  the  radical  sign 
or  fractional  index. 

We  have  already  seen  (Art.  290)  that  a  part  of  the  root  may 
be  removed  from  under  radicals  of  the  form  Va'^b,     Thus, 
^~^=  Va^Xb=  V^x  Vb=aV'^. 
Now,  by  reversing  this  process, 

a Vb=z ;/T" X  !t/b=  ^~cFxb=  yo^. 
Hence  we  have  the  following 

RULE. 

1.  Raise  the  rational  coefficient  to  a  power  of  the  same  name 
as  the  root  indicated  by  the  radical  sign  or  fractional  index. 


186  ELEMENTS    OF   ALGEBRA.  [SECT.  VI. 

2.  Multiply  the  quantity  under  the  radical  sign  or  index  by 
this  power,  and  place  the  given  radical  sign  or  index  over  the 
product, 

EXAMPLES. 

1.  In  the  expression  3a\/26,  let  the  coefficient  be  introdu- 
ced under  the  radical. 

Za^/2b=: \/~9? X  y/2h=:'^Wx2bz=  >/l8a^.     Ans. 

2.  In  the  expression  ia^-^Sax,  let  the  coefficient  be  intro- 
duced under  the  radicaL  ^ns.  y/l92a'^x. 

3.  In  the  expression  2a\4:Xyy,  let  the  coefficient  be  intro- 
duced under  the  fractional  index. 

2a\4xy)^=(8ay  x  {4xyy  =  (Sa'  X  4xyY={32a^xyY.     Ans, 

4.  In  the  expression  6v^l3,  let  the  coefficient  be  introdu- 
ced under  the  radical.  .  Ans.  \/2808. 

5.  In  the  expression  2al{2db^y,  let  the  rational  cofficient 

be  introduced  under  the  fractional  index.       Ans.  (IQa^h^)  . 

6.  In  the  expression  (a +  Z>)\/aA,  let  the  rational  coefficient 
be  introduced  under  the  radical  sign.  

Ans.  Vl^'Wxab='sf a%^2a^y'-\-ah\ 

7.  In  the  expression  ^  (__£_)  ,  let  the  rational  coeffi- 
cient be  introduced  under  the  fractional  index. 

Ans,   '    "*"  ^    ' 


\a%''-\-h'') 


CASE  III. 

299.  To  reduce  radicals  of  different  indices  to  equivalent  radi- 
cals having  a  common  fractional  index. 

RULE. 

1.  Reduce  the  indices  to  a  common  denominator. 

2.  Involve  each  quantity  to  the  power  expressed  by  the  numer- 
ator  of  the  reduced  index. 

3.  Take  the  root  denoted  by  the  denominator, 

EXAMPLES. 
1  1 

1.  Reduce  a^  and  b^  to  a  common  index. 

a*=a'^^^=(a''y^,  i  i 

3      ^   \         Ans,  (a^y^,  and  (b^y^ 
b^z=b^^=(bY^, 


SECT.  VI.]  ADDITION    OF   RADICALS.  187 

2.  Reduce  2    and  3    to  a  common  index. 

_  _  Ans.  8*^,  and  9^» 

3.  Reduce  \/^  and  v^^  to  a  common  index. 

vT=(i)*=(i)'=((i)')*=(TV)*- 

4..  Reduce  (  ^  ]    and  |  ^  ]     to  a  common  index. 

_  ^~.(C)'-(|)'. 

5.  Reduce  v^|,  y/\y  and  ^2  to  a  common  index. 

__  Ans.  i'^l296,  ^'V656T,  and  '-^8. 

6.  Reduce  V^h  ^"^^  v'5^  to  a  common  index. 

Am.  V  150^j,  and  '^151^^ 

CASE   IV. 
ADDITION  OP  RADICALS. 

300.  If  the  radicals  are  not  similar,  and  cannot  be  made 
80  by  reduction,  it  is  evident  that  the  addition  can  only  be 
expressed. 

Thus,  s/a-\->/h  can  be  reduced  to  no  simpler  form. 

301.  If  the  radicals  are  similar,  they  may  be  added  by  the 
following' 

RULE. 

Add  the  coefficients^  and  to  their  sum  annex  the  common  radical, 
J^ote. — If  the  radicals  are  not  similar,  they  may  frequently 
be  made  so  by  reduction. 

EXAMPLES. 

1.  Add  4\/ax,  2\/aj:,  5v^ax,  and  Z^^/ax^ 

^\/ax 
2s/ax 
by/ax 
3iv'ai 
lii'/ox.  Ans.  14iJ\/ax. 

2.  Add  3a^^ and  5c^^  Ans.  (3a+5c)^|. 


.J^»mt 


188  ELEMENTS    OF    ALGEBRA. 

3.  Add\/8andN/32. 


[sect.  VI. 


v/8  =V4>   x2==v/4   X\/2=:2v/2. 


V32=n/16x2=x/16x%/2-4v/2. 


4.  Add  i/50  and  v/l28. 

5.  Add  (SGa^y)^  and  (25y)^. 

6.  Add  ^54^  and  ^1280^. 


Jlns.  eV2. 

Ans.  13\/2. 
'  Ans.  (6aH-5)v/y. 

Ans.  4av/J-y. 


7.  Addy^^^d  y/^. 

/Q>,2  /9^2 


s/ 


Va'   Xj\  =  Va'   XVj\=:  «n/tV- 


4a\/yV- 
21 

24 
Ans.  (b-^y)Vb^^ 
Ans.  4<a^2b. 


8.Add./^and,/122. 
V  147  V  294 

9.  Add^i"and\V^^. 

'  V  13824 

10.  Add  i/b^  and  Vby^ 

11.  Add  ^"32^  and  2a</2b. 

12.  Add  4>(a+x)^  and  (4a'^>2+4a%)^. 

./?»s.  (4+2a5)v/a4-a7» 

13.  Add  v/f",  4v'l2,  and  3  s/~J~  Ans.  9v/3. 

14.  Add  ^192  and  ^24.  Ans.  6^3. 

15.  Add  3^5^16^  and  aby^^.  Ans.  9ab^^. 

CASE    V. 
SUBTRACTION  OF  RADICALS. 

302.  If  the  radicals  are  not  similar,  and  cannot  be  made 
so  by  reduction,  the  subtraction  can  only  be  expressed. 

Thus,  y/a — Vb  can  be  reduced  to  no  simpler  form. 

303.  If  the  radicals  are  similar,  the  subtraction  may  be 
performed  by  the  following 


SECT.  VI.]  MULTIPLICATION    OF    RADICALS.  189 

RULE. 

Subtract  the  coefficient  of  the  subtrahend  from  the  coefficient  of 
the  minuend,  and  to  the  difference  annex  the  common  radical, 

J^ote. — If  the  radicals  are  not  similar,  they  may  frequently 
be  made  so  by  reduction. 

EXAMPLES. 

1.  From  ^ab^/cd  subtract  SabVcd,  Ans,  aby/cd. 


2.  From  V^Oa^  subtract  VSa\ 


V50a^=v/25a^  X  2=i/25a^  x  V2=baV2. 


y/Sa'  =s/4>a'   x2=v/4a'   Xv/2=2av/2. 


3aV2.    Jlns, 
3.  From  -5^192  subtract  4^24-.  jJns.  2^3. 


4.  From  IW^Qa^bc"  subtract  3av/24^(r'.    JJns.  3Sacs^6bc, 

5.  From  v/|  subtract  \/^.  JJns.  j^n/I^. 

6.  From  Vi  subtract  v/|.  ^ns.  Jv^3. 

7.  From  3^^  subtract  1^400.  JJns.  J^^Bo. 

8.  From  4a^l250a^  subtract  i^640a«.  Ans.  ISaVlO. 

9.  From  1^^567  subtract  1^112.  Ans.  -^^1, 

10.  From  ^^%W  subtract  fv/Qo'.  Am,  |a. 

CASE    VI. 

MULTIPLICATION  OF  RADICALS. 
604.  If  the  quantities  are  not  under  the  same  radical  sign, 
and  are  not  roots  of  the  same  letters,  the  multiplication  can 
only  be  indicated.  But  since  the  product  of  the  roots  is 
equal  to  the  root  of  the  product  (Art.  156),  and  since  the 
product  of  several  factors,  composed  of  the  same  letters  or 
quantities,  is  obtained  by  taking  one  of  the  factors  affected 
by  an  exponent  equal  to  the  sum  of  the  exponents  of  the 

several  factors  (Art.    88),  that  is,  a"*  x  a** =a'*»x  «***=«"", 
we  have  the  following  general 


190  ELEMENTS  OF  ALGEBRA.  [SECT.  VI. 

RULE. 

1.  If  the,  quantities  are  under  the  same  radical  sign  or  index, 
multiply  thefn  like  rational  quantities,  and  place  the  common 
radical  sign  over  the  product. 

2.  If  the  qu(fitities  are  composed  of  the  same  letters  or  num^ 
bers,  and  are  affected  with  different  fractional  exponents;  add 
these  exponents. 

3.  If  the  radicals  have  rational  coefficients,  multiply  them,  and 
prefix  the  product  to  the  product  of  the  radicals. 

EXAMPLES. 


1.  Multiply  2ay/3ax  by  SbVSa^cx. 


2a>/3ax  X  Sb\/3a^cx=Qaby/3ax  x  3a^cxz=6ab\/^ci^ca^' 


6aby/9a'^cx^=z6aby/9a^x^  X  ac=6a5\/9aV  X  y/ac=6ab  x  Sax^ac 
=  ISc^bxy/ac,     Arts. 

%  Multiply  {3af  by  (3a)*.  Ans.  ^~^Ma^. 

{3af=:{3af 

{3af={3ay 


5  1 


(3ay=(24^3a'y=z  ^24<3a'. 

3.  Multiply  4n/2  by  V6.  Ans.  4.5^288. 

4.  Multiply  5v/5  by  3v^8.  Ans.  30x/10. 

5.  Multiply  2av/aM^  by  ^3aV^Tb^ 

^  Ans,  —ea'ia'-i-b'). 

6.  Multiply  5a^a+x  by  UV(a~+xf.       Ans.  20ab{a-\-x). 

8.  Multiply  (6a^bcy  by  (6a'bc)K  Ans.  (6a'bcyK 

9.  Multiply  a  ,  a  ,  and  a   together.  Ans.  a^^. 

10.  Multiply  7e^l8  by  5^4.  Ans.  70^9. 

11.  Multiply  2a%a+bY  by  6ac(a-\-by. 

Ans.  12a^c(a-{-b)~i^ . 

12.  Multiply  6aV3a^  by  64^27a^  Ans.  dOa'Vb. 

13.  Multiply  4>^J^  by  3s/8.  Ans.  12^2. 

14.  Multiply  27  V  by  iVh  ^'^^'  ^^^Vt- 


8£CT.  TX.]                     DIVISION   OF  RADICALS. 

IS 

15.  Multiply  4.^f  by  ia^J 

•*"! 

16.  Multiply  a¥  by  flM. 

\dn8,ab^. 

17.  Multiply  3a-V  by  3a**~«. 

Am.  9. 

18.  Multiply  laJ"  by  llxK 

3    1                  1 

^715.  14a*x^. 

.        '3      3 

19.  Multiply  6a V  by  1  la^a:.  Ans,  %d^  £^ . 

CASE   VII. 

DIVISION  OF  RADICALS. 

305.  If  the  numbers  are  not  under  the  same  radical  sign, 
nor  roots  of  the  same  letters,  the  division  can  only  be  indi- 
cated. 

306.  But  in  those  two  cases  it  may  be  performed  by  the 
following  general 

RULE. 

1.  Jf  the  quantities  are  under  the  same  radical  sign  or  index, 
divide  them  like  rational  quantities,  and  place  the  common  radi- 
cal sign  over  the  quotient. 

2.  If  the  quantities  are  composed  of  the  same  letters  or  num- 
bers, and  affected  vrith  different  fractional  exponents,  subtract  the 
exponent  of  the  divisor  from  that  of  the  dividend. 

3.  If  the  radicals  have  rational  coefficients,  divide  the  coeffi- 
cient in  the  dividend  by -that  in  the  divisor. 

EXAMPLES. 


1.  Divide  6v/12a^Z>  by  3v/3a*.  Ans.  Wa. 

V     oah 

2.  Divide  Ua^  by  7a^'  Ans.  2a^ 

14a'-^7a*  =  14a^~7a^=2a^-^=2a'  ; 
Or,  14^^H-7^/a=14y^-^7W=2e^a. 

3.  Divide  4*  by  4^.  Ans.  4^ 

4.  Divide  6v/54  by  3v/2.  Ans.  6v/3. 

5.  Divide  4^72  by  2>/18.  Ans.  Wl> 


192  ELEMENTS    OF   ALGEBRA.  [sECT.  VI. 


6.  Divide  v/7  by  V7. 

^;js.  V7. 

7.  Divide  8v/108  by  276. 

Ans.  12n/2. 

8.  Divide  (a%H'f  by  /. 

^»s.  (a&f . 

9.  Divide  v/3  by  v/|. 

.^W5.  v/|,  or  ^v/,6. 

10.  Divide  i^J  by  i^I. 

^7i5.  f  4^l2. 

11.  Divide  %ah^hy1ah^. 

Ans.  A^a^h^, 

12.  Divide  21a&'^'  bv  3a*A 

Am.  7a¥. 

CASE   VIII. 

INVOLUTION  OF  RADICALS. 

307.  Let  it  be  required  to  involve  ^as/W  to  the  second 
power.  By  the  definition  of  involution  (Art.  25),  we  shall 
have  (6a\/35^)^=6aN/36^  X  6a\/36^ ;  or,  performing  the  multi- 
plication, (6av/3^f = 6ax/3^^  X  6av/"36^= 36aV"96'^= 36a=^  X  3&^ 
=  108a2&^ 

\    3 

Again,  let  it  be  required  to  involve  3a  h^  to  the  third 
power. 

(3a¥)='z:r3aVx  3a^^»^X  3a¥=27a^6l 

308.  The  above  operation  will  evidently  apply  to  all  cases 
of  monomial  radicals. 

Hence  we  have  the  following  general 

RULE. 

1.  Involve  the  coefficients  to  the  required  power, 

2.  If  the  number  is  under  the  radical  sign,  involve  it  as  if  it 
were  rational ;  over  the  power  place  the  radical  sign,  and  then 
reduce  the  result  to  its  simplest  form. 

3.  If  the  number  to  be  involved  is  affected  by  a  fractional 
exponent^  multiply  the  exponent  of  each  letter  by  the  index  of  the 
required  power. 

EXAMPLES. 

1.  Required  the  third  power  of  ^a^y.  Ans.  'Ula^y. 

2.  Required  the  second  power  of  4a^6\/6&. 

Ans.  16a%V36F'=96a3&2. 


.#--^- 


SECT.  VI.]  EVOLUTION    OF    RADICALS.  198 

3     1        

3.  Required  the  third  power  of  ^'6^\/86*x. 

^ns,  1288a«6V2i. 

4.  Required  the  fourth  power  of  6\/^.  ^ns.  36. 

5.  Required  the  third  power  of  b^2a^b, 

Jlns.  125a^8a^. 

6.  Required  the  third  power  of  3%/?  x  2^2. 

Jlns,  216>/J. 

CASE  IX. 
EVOLUTION  OF  RADICALS. 
309.  From  the  foregoing  operations,  the  following  rule 
win  be  sufficiently  obvious  : 

RULE. 

i.  Extract  the  root  of  the  coefficient^  if  it  is  a  complete  power ; 
if  not,  introduce  it  under  the  radical  sign  or  index. 

2.'  If  the  radical  sign  is  used,  multiply  the  figure  over  the  foot 
of  the  radical  by  the  index  denoting  the  root  to  he  taken, 

3.  If  the  fractional  index  is  used,  divide  the  index  of  each  let' 
ter  by  the  index  of  the  required  root. 

EXAMPLES. 

1.  Extract  the  square  root  of  IGa'^V^o:". 

2.  Extract  the  third  root  of  27a«^26t/.      JJns.  3a^  V^bd. 

3.  Extract  the  third  root  of  8oM.  Jlns.  2a^b\ 

4.  Extract  the  third  root  of  6U^VTW.  Jlns.  i^a^^iW. 


5.  Extract  the  square  root  of  24^3a.         Jlns,  2yi08a. 

6.  Extract  the  cube  root  of  54aVVi.      Jlns,  Sa^b^^Q. 

CASE  X. 

POLYNOMIALS  HAVING  RADICAL  TERMS. 
310.  We  will  give,  for  the  exercise  of  the  learner,  some 
examples  of  polynomials  having  one  or  more  of  their  terms 
radical  quantities.     These  examples  may  be  solved  by  an 
application  of  the  rules  laid  down  in  the  preceding  cases, 
n  Bb  ) 


194  ELEMENTS    OF   ALGEBRA.  [SECT.  VI. 

EXAMPLES. 

1.  Required  the  second  power  of  a-\-Vy* 

Ans,  a^-\-2a>/y+y, 

2.  Required  the  third  power  of  a — Vh» 

Am.  (^—^aW~b-{-^ah—y/¥. 

3.  Required  the  second  power  of  \/3a4-\/2a7. 

Ans.  3a+\/6aa;4-2ar. 

4.  Required  the  square  root  of  a-\-2^/ab-\-h, 

Ans.  y/a+y/b. 

5.  Required  the  square  root  of  9a+36\/Sax-{-10Sx. 

Ans.  3Va-\-6V3cc. 

6.  Required  the  cube  root  of  (f+3a''^'x-\-3aVoo^-x. 

Ans,  a-\-^x. 

CASE    XI. 

BINOMIAL  AND  TRINOMIAL  SURDS. 
311.  Expressions  under  this  form,  y/a-\-Vhj  or  a-\-Vi,  are 
called  binomial  surds,  and  may  be  reduced  to  rational  quan- 
tities on  the  principle  that  the  product  of  the  sum  and  differ- 
ence of  two  quantities  is  equal  to  the  difference  of  their  squares, 
.  Thus  the  binomial  surd  \/a-\-\/b 
Multiplied  by  -         -       y/a—s/h 


uW 


a-\-y/ab 
—/ah^l) 


rives       -         -         -  a  +^j  a  rational  quantity. 

312.  Trinomial  surds  may  be  reduced,  first,  to  binomial 
surds,  then  to  rational  quantities.     Thus, 


The  trinomial  surd     - 

,Ja-\-^fh  —^fc 

Multiplied  by     - 

yfa—s/h   +n/c 

. 

a-\-y/al — \/ac 

—y/ab           —b-\-   Vbc 

-hVac       +   Vbc — c 

Gives 

a                     ^b-i-2Vbc—c. 

SECT.  VI.]  BINOMIAL  AND  TRINOMIAL  SURDS. 


191 


A 


Let  x=a — b — c  /  then  we  shall  have 

Multiplying  by     -         -       x  — 2>/bc 

x*-^2xVTc 
^  — 2xy/bc — ifbc 

x*  — 46c 

Restoring  value  of  x  -  (a — b — cf — ibc. 
3.  Find  a  factor  which  will  make  1  +  v/2  rational. 

l  +  %/2 

1— v/2 


l  +  v^2 
— v/2— 2 


1  — 2=-l. 

Hence  the  factor  is  1 — \/2. 

4.  Find  a  factor  which  will  make  n/10 — V2 — \/3  rational. 
VT6—V2  -V3 

Multiplying  by   -   x/'T0-hv'2  4-v/3 

10— v/20— v/30 
— 2+N/20  — /6 

—3  +'/30— v^6 


Multiplying  by 


—  2n/6 
-f  2v'6 


25 


— lOv/6 
4-10v/6-24 


25  —24=1. 

Hence  the  factors  are  v/'io+v/24-v/3,  and  5  +  2v/6. 

5.  Find  a  factor  which  will  make  3— 2n/2  rational. 

6.  Find  a  factor  which  will  make  \/6  +  3v/2 — v/5  rational. 
313.  By  the  above  process,  fractions  may  be  cleared  from 


196  ELEMENTS    OF    ALGEBRA.  [SECT.  VI. 

radical  numerators  or  denominators  without  altering  the 
value  of  the  fraction,  and  thus  the  process  of  extracting  the 
root  be  facilitated  by  confining  it  either  to  the  numerator  or 
denominator. 

1.  Let  it  be  required  to  extract  the  square  root  of  the 


fraction  ^. 


a     \/a     y/a  x  y/a> 


Vb     \fh  X  \/a     \/ab 

2.  Extract  the  square  root  of  the  fraction  ^      . 

ooy 

a-{-b_\/a-\-b     s/a-\-bXy/a-\-b  a-\-b 


4- 


^y        Vxy         Vxy-{-Va-^b       ^/axy-\-bxy 


3.  Extract  the  square  root  of  -. 


V- 


5_n/5_n/5xn/5^V25_     5 

8     v/8     n/8x\/5~x/40~2n/10' 


\/2 

4.  Reduce  the  fraction =  to  an  equivalent  fraction 

3— v/2  ^ 


having  a  rational  denominator. 


Ans^J^^. 


3 

5.  Reduce  the  fraction  — = =to  an  equivalent  fraction 

having  a  rational  denominator. 

..  .  Ans.^^^1. 

1 

6 

6.  Reduce  the  fraction  ~t  to  an  equivalent  fraction  hav- 

5* 

ing  a  rational  denominator. 

5 
g 

7.  Reduce  the  fraction  -^ = to  an  equivalent  frac- 

v/3+v^2+l 

tion  having  a  rational  denominator. 

*/??i5.  4— 2V6  +  2v'2. 


SECT.  VI.]  BINOMIAL  AND  TRINOMIAL  SURDS.  1^ 

8.  Reduce  the  fraction  - —    v   ^-hv     ^^  ^^  equivalent 
fraction  having  a  rational  numerator. 


84v/10-20>/64-72v/34-20W5 

I 

CASE    XII. 

ROOTS  OP  BINOMIAL  SURDS  OF  THE  FORM  a±^'b. 
314.  It  is  proposed  to  obtain  a  formula  for  extracting  the 
square  roots  of  expressions  in  the  form  of  a±  y/b. 

Let         -         ya-\-Jl=x+y/~y  (1). 

Then      -         y  a-^/b=x-'^/y  (2). 

Squaring  both  equations, 

a+>/^=«2+2a:v/y4-y        (3). 

a— %/^=x*— 2xv/y-hy  (4). 
Adding       -      2a       =2a:*  +2y      (5). 

And     -      -  a=oi^-\-y  (6). 

Multiplying  the  first  equation  by  the  second, 

^~^^~b=3?-y.  (7). 

Adding  the  sixth  and  seventh  equations, 

a^shr^h^lx"  (8). 

Reducing   -     a'=y/°+^^'~^  (9). 

Subtracting  the  seventh  from  the  sixth  equations^ 
a-^ir^z=:'Xy  (10). 

Reducing    -     v^y^-y/^LZ^J^  (l^)' 

Substituting  these  values  of  x  and  >/y  in  the  first  and  sec- 
ond equations, 


\/a+V*=v/^^^*V^^ 


-y/a^^b 


Va-x/6=V 2 V 2 


198  ELEMENTS    OF   ALGEBRA.  [sECT.  TI. 

Or,  letting  c=y/a^ — b. 

1-    -   ^^^=^/^+\/^    (A). 

EXAMPLES. 

1.  Extract  the  square  root  of  34-2'/2. 

Here  a=3,  andv7z>z=2v/~2=rv/8,  or  h—^,  and  c^-ZO— 8=1. 

3+2n/2=34-v/8 


\/3+2v/2=n/2+1.     ^715. 

2.  Extract  the  square  root  of  14— 6V5.       Ans.  3 — \/5. 

3.  Extract  the  square  root  of  ll  +  6\/2.       Ans.  3+-/2. 

4.  Extract  the  square  root  of  7 — 2\/10.    Ans.  V5 — v/2. 

5.  Extract  the  square  root  of  94H-42n/5.  ^7^5.  7+3\/5. 

6.  Extract  the  square  root  of  1+W — 3. 

Ans,  2+1/^. 

7.  Extract  the  square  root  of  28+1073.     Ans.  5  +  n/3. 

8.  Extract  the  square  root  of  ab-\- 4)6^— d^-\-2\/^ahc^ — abd. 

Ans.  V'ab+V^c'—d^' 

9.  Find  the  sum  of  y  16  +  30v/=^+\/l6— 30%/=!. 

Ans.  10. 

10.  Find  the  sum  of  \/bc-\-2bVbc—b'-\-  ybc—2b^bc^b\ 

Ans.  26 


^ 


SECT.  YII.]   EQUATIONS  EXCEEDING  THE  FIRST  DECREE.       199 


SECTION  VII. 
Equations  of  the  Second  Degree, 
EaUATIONS  EXCEEDING  THE  FIRST  DEGREE. 

315.  The  questions  heretofore  discussed  involved  only 
the  first  power  of  the  unknown  quantity,  or,  if  a  higher 
power  ever  appeared,  it  was  cancelled  in  the  process  of  the 
reduction.  The  enunciation  of  other  questions,  however, 
frequently  requires  a  power  or  root  of  the  unknown  quantity, 
and  for  the  solution  of  such  cases  we  must  seek  for  meth- 
ods different  from  any  heretofore  discussed. 

316.  Equations  of  this  nature  are  divided  into  two  class- 
es, viz.,  pure  or  incomplete  equations,  and  affected  or  complete 
equations. 

317.  A  pure  equation  is  one  which,  when  reduced  to  its 
simplest  form,  involves  only  one  power  or  root  of  the  unknown 
quantity.     Thus, 

X  =q  is  a  pure  equation  of  the  first  degree  j 

a^=q^  is  a  pure  equation  of  the  second  degree,  or  a  pure 

quadratic  equation; 
s?=:(f  is  a  pure  equation  of  the  third  degree,  or  a  pure 

cubic  equation  ; 
x*=q*  is  a  pure  equation  of  the  fourth  degree,  or  a  pure 

biquadratic  equation,  &c. 

y/Xy  or  x^=q^y  is  a  sub-quadratic  equation ; 

—  L       1    . 

^Xf  or  x^=q^,  is  a  sub-cubic  equation  ; 

V«,  or  X*  =q*,  is  a  sub-biquadratic  equation,  &c. 

318.  An  affected  equation  is  one  which,  when  reduced  to  its 
simplest  form,  involves  different  powers  or  roots  of  the  un- 
known quantity.     Thus, 

a^-{-px=q  is  an  afl!ected  quadratic  equation  ; 
a'-f-JP*-fpJC=fl  is  an  afi^ected  cubic  equation,  &c.       ' 


200  ELEMENTS    OP   ALGEBRA.  [SECT.  VII. 

PURE  EaUATIONS. 

319.  Pure  equations  may  be  readily  solved  on  the  princi- 
ples, 1.  If  the  same  root  of  loth  mernbers  of  an  equation  be  eX' 
traded,  the  results  mil  be  equal.  2.  If  both  members  be  involv- 
ed to  the  same  power,  the  results  will  be  equal  (Art.  158). 

Let  us  take  the  equation  -         -         -         x^^^q^ ; 

Extracting  the  nth  root   -        -        -        -        x  =q. 

Again,  the  equation  -         -         -         -     i^x  —  V q  ; 

Involving  to  the  ?2th  power      -        -        -         a?  =:j. 

Hence,  for  the  reduction  of  pure  equations,  we  have  the 
following  general 

RULE. 

1.  Reduce  the  equation  to  such  a  form  that  the  power  or  root 
of  the  unknown  quantity  may  stand  by  itself  in  the  first,  and  the 
known  quantity  by  itself  in  the  second  member  of  the  equation. 

2.  If  the  expression  containing  the  unknown  quantity  is  a 
power,  extract  the  corresponding  root  of  both  members. 

3.  If  the  expression  containing  the  unknown  quantity  is  a 
root,  involve  both  members  to  a  power  of  the  same  name.  .^^ 

J^ote  1. — If  the  even  root,  i.  e.,  the  second,  fourth,  or  sixth 
root  of  an  equation,  is  to  be  taken,  the  resulting  second 
member  should  be  affected  by  the  double  sign  ± .  Thus^ 
the  square  root  o{  q^—:kq;  for  -\-qx-\-q—+q^,  and — qx  —q 
= +^' :  the  fourth  root  of  q'^=  it g ;  (or  qxqxqxq=q'^i  and 
— 5'X — qX—qX — q=-{-q'^:  and  the  sixth  root  of  q^=dzq  ; 
for  q-q-q'q'q'q=q\  and  — qX—qX — qx—qX—qX—q=q\ 
&c. 

JSTote  2. — Since,  if  the  even  root  be  taken  on  both  sides  of 
the  equation,  it  would  be  very  natural  to  suppose  that  the 
first  member,  or  x,  should  be  affected  with  the  double  sign 
± ,  as  well  as  the  second  member  of  the  equation.  Affect- 
ing it  thus,  and  arranging  the  signs  in  the  equation,  ±x= 
±  q,  in  every  possible  manner,  we  shall  have  the  four  equa- 
tions. 


(1).     +0?=+?. 


(3).     -x=+q. 


(2).     -\-oo=—q.  (4).     •^x=-~q. 


^ 

••".^\^ 


SECT.  VII.]  PURE    EQUATIONS.  201 

But  Still  we  have  in  reality  no  more  than  the  first  two 
equations,  as  the  third  equation  expresses  the  same  rela- 
tions with  the  second ;  for,  changing  the  signs,  it  becomes 
(2).  -\-x=—q  : 

And  the  fourth  expresses  the  same  relations  with  the  first ; 
for,  changing  the  signs,  it  becomes 

(1).         -^x=-\-q. 

EXAMPLES. 

1.  Find  the  value  of  x  in  the  equation  — — 10=2» 

4 

2^—10=  2; 

Clearing  of  fractions        -        -        3a?* — 40=  8  ; 
Transposing  and  reducing       -         3x*         =48 ; 

Dividing a?*         =16; 

Evolving  -        -        -        -  X         =±4. 

Verifying  on  the  supposition  that  x=4-4  ; 

2:1^—10= 12—10=2. 

4 

Verifying  on  the  supposition  that  a:=— 4  ; 

?2ir±'~10=2ll^=12— 10=2. 
4  4 


2.  Find  the  value  of  ar  in  the  equation  >/x — 16=8 — %/a?. 

-y/x — 16=8 — Vx  ; 
Involving  both  members    -    x — 16=64 — 16v^a:-l-ap; 
Cancelling  and  transposing      16\/a:=80 ; 
Dividing    -        -        -        -         y/x=  5 ; 
Involving  -        -        -        -  x=25. 

Verification        -        -        V25^6=8--n/25  ; 
Or v/9=8— 5  =  3. 

X — ax     y/x 

3,  Find  the  value  of  x  in  the  equation  — ;=—  =  — . 

VX  X 

Clearing  the  equation  of  fractions    -     x' — ax^=x  ,* 

Dividing  by  X x — ax=l;^ 

Besolving  into  factors      ...  (1 — a).x=l ; 
Cc 


202  ELEMENTS    OF   ALGEBRA.  [sECT.  VII. 

Dividing  by  the  coefficient  of  a?        -  a?= 


4.  Find  the  value  of  a?  in  the  equation  Vx-j-a-. 


1—a 

a+b 

Voc — a 

Clearing  of  fractions        -       Vx^ — a^=  a+b  ; 
Involving  ...  oc^ — a^=  a^-\-2ab-\-bf^ ; 

Transposing  and  reducing        -        x^='2,a^-\-2ab-\-b^ ; 


Evolving  ....        X  =±V2a'—2ab-{-b^. 

.       \/^+28     v/a:+38 

5.  Find  the  value  of  x  in  the  equation  — -= — -  =—rz — — , 

\/x-{-4f      \/a?+6 

Clearing  of  fractions   -    a?4-34%/^H-168=a?+42\/a?+152; 
Transposing        .         -         -         -     8y/x=  16 ; 
Dividing     -----       Vx—  2j 
Involving    -        -        -        -        -  x=  4f. 

.  .      Vax — b     3s/ax — 25 

6.  Find  the  value  of  x  in  the  equation  —= — -:=—-== — — . 

Vax-\-b     3\/aa?+56 

Clearing  of  fractions, 

3ax-{-2bV'ax—6b^=3ax-{-bVax^^b^ ; 
Transposing        -         -        -      bs/ax=z3b^; 
Dividing  by  &      -         -         -        ^/ax=z3b; 
Involving    -        -        -        -  ax=9b^ ; 

Dividing    -        -        .        -  x= — . 


7.   Find  the  value  of  x  in  the  equation  \/ x+Vco— 

V  ^         2  V    x-\-Vx  _ 

/ z  _       3v/f. 

Multiplying  by \/  a:+  v/a?    -  x+Vx-^s/a^ — a?=— 2~  ' 


\/a? 


Transposing  and  reducing    -        -  x — ---=y/a^ — x; 

At 


Dividing  by  Va;    -         -         -         -  Va? — \z=L\fx — 1 ; 

Involving x — s/x-^\—x — 1 ; 

Transposing  and  reducing    -        -        -        '>Jx=i\  j 
Involving 


X  — 


SECT.  VII.]  PURE    EQUATIONS.  S08 

!*       20" 

8.  Find  the  value  of  x  in  the  equation  x-\->/a*-\-3^:=z—rT==z 

Clearing  of  fractions         -  Xy/c^-\-3^-\-  (^-\-a^z=2a?-y 

Transposing  and  reducing,  Xy/a^-\-a^=2(^ — a* — x*=a^ — «•  j 

Involving  -        -  2c'(a'-{-a^)=  a*— 2aV4-a?<j 

Multiplying  factors  -        -     c^a^-}-x*=:  a* — 2aV+iC* ;  *^ 

Transposing  and  reducing         3aV=  a*  j  » 

a  — 

Dividing  and  evolving      -        -     «  =~75»  **'  ^v^i* 

v3 

ji 

9.  Find  the  value  of  x  in  the  equation  3a?»— 29=:_4-510. 

4 

^;w.  a?=  14. 


10.  Find  the  value  of  a;  in  the  equation  \/a:— 32=\/x — ^ 
\/32.  ./f;i5.ar=50. 

/20x' 9        - 

11.  Find  the  value  of  x  in  the  equation  W =:^/x, 

V     4>x 

Ans,  x=\. 


12.  Find  the  value  of  x  in  the  equation  \/x+v/3+a?= 
6 

Ans.  a:=l. 


13.  Find  the  value  of  x  in  the  equation  x-\-\/W^\^— 
-^.  An..  x=y3. 

14.  Find  the  value  of  x  in  the  equation 

ap+2=  \/4H-«x/64+^.  Ans.  x^^. 


15.  Find  the  value  of  x  in  the  equation  \/2ar'+9x*-f  27x 
=a:+3.  w^TW.  x=3. 


16.  Find  the  value  of  a;  in  the  equation  ^/x — 32=16 — s/x. 

Ans,  x=%\. 
v/6x-2    4n/6x~9 


17.  Find  the  value  of  x  in  the  equation 


>/6a:-f-2     4N/6ar+6 
Ans.  x=6. 


18.  Find  the  value  of  x  in  the  equation  ^x^— a'=  4^3ax« — 


3a'x-f86.  Ans.x=a^-Wb. 


204  ELEMENTS    OF  ALGEBRA.  [sECT.  VII, 

19.  Find  the  value  of  a?  in  the  equation  c? — 2aa7+a?2=:Z>. 

Ans,  x=a^^>/b. 

a — s/d^ — a^ 

20.  Find  the  value  of  x  in  the  equation —  &. 

^  a+v/a^— a?2 

Multiplying  both  numerator  and  denominator  of  the  first 


member  by  a — \/a^ — x^. 


Reducing  and  clearing  of 
fractions  -        -         - 


(a— vV— ^_, 


Evolving     - 
Transposing 
Involving     -         -         - 
Cancelling  and  transposing 
Dividing  by  a?      - 
Re  solving-  into  factors 


Dividing 


-  a—Va^ — a^=±xy/b; 
a  q=  X\/l— y/a^ — x^ ; 
a^  qp  2aa?x/5"+  bx^ = a^—x? ; 

bx'^-x^—±1axsfb; 
bx-\-x^±1a^fb', 


X—. 


^'+1 


21.  Find  the  value  of  x  in  the  equation 


\/x-\-\/x — a 


Vx — \/a? — a    X — a 
Multiplying  the  numerator  and  denominator  of  the  first 
fraction  by  Vx — Vx — a. 


Whence 


Evolving 


Clearing  of  fractions 


X— 

ix-a) 

n'a 

(Vx- 

Wx—af 
1 

X — a 

(Vx- 

Wx-af 
1 

~x — a 
±n 

Vx 

— Vx — a 

Vx — a 

Multiplying  by  Voc — a 

Or  .        -         - 

Transposing  -      (x — a)-\-n{x — a)=:±n\/!X^ — ax; 


s/x — a=  ±  »(\/aj — \/x — a)  ; 
X — a=  ±  niVx^ — ax — x-\-a), 
X — a=  ±  nVoc^ — ax — n{x — a)  y 


SECT.  VII.]  PURE    EQUATIONS.  206 

Resolvinpf   into  )     *■»  ,     \  ,         \       ,       /Ta 

°  [    (1-l-n).  (x — a)=±n>/a^ — ax; 

factors        -    > 

InTolving        -       (1-fn)'.  (x — af=n\3p^ — ex)  =ft'x(x— a)  ; 

Dividing  by  X — a,  (1-fw)^  (x — o)=«'x, 

Or  -.        (l-\-nyx—{\—nfa=n^x; 

Transposing  -  (l+n)'x — n*x=(l-|-n)^a, 

Or  -  (l  +  2;i-|-n')x— »"x=(14-»)*a; 

Addingcoeffi.;    ^^2^^^,_^,^^^  ^^^^,    . 

cients  of  X  > 

Reducing        -         -        (l  +  2;i)x=(l+n)^a  ; 

Dividing  by  l+2;i  -  x^llt^.     ^;w. 


_                                                            \/a-|-x4-\/a — X 
22.  Find  the  value  of  x  in  the  equation  -^ =6. 


23.  Find  the  value  of  x  in  the  equation 
1  n/3 

24.  Find  the  value  of  x  in  the  equation  \/x'+8=v/125 


— 6x*— 12x.  ^»5.  x=3. 

25.  Find  the  value  of  x  in  the  equation*  /.^il5-}- 2*  /    ^ 

V       X  V    a-i-a: 


-V. 


.^TW.  X= — .. 


Examples  of  Pure  Equations  containing  two  or  more  unknown, 
Quantities. 
320.  In  equations  of  this  kind,  unknown  quantities  may 
be  eliminated  by  the  same  principles  that  were  applied  in 
equations  of  the  first  degree. 

EXAMPLES. 

1.  Find  the  values  of  x  and  y  in  the  equations  x'+y=28, 

and  ^—^=19.  ^ns,  x=5,  and  y=3» 

5       3 

18 


206  ELEMENTS   OP    ALGEBRA.  [sECl^.  VII 

5       3 

Transposing  y,  and  extract-  ) 

ing  the  square  root  of  the  }              a?=  ^/28 — y 
first  equation  -         -         -  J 

Clearing  the   second  equa-  }    ^  „     ^ 

tion  of  fractions      -         J  12a:^-5y=285 

Transposing,  &c.         .         -        -       x^=?^^±^ 


Evolving x=a/ 


285 -f5y 
12 
Forming  a  new  equation  of  the  two  values  of  x, 


28-y=5?^±^ 
^  12 

336— 122/=285+5y 

— 12y— 52/=285— 336 

172/=51 

y=3,  and  a:=5. 

2.  Find  the  values  of  x  and  y  in  the  equations  3y=aj+y, 

and  xy=  18.  ^?i.9.  3?=  db  6,  and  y=  dr  3. 

*  3.  Find  the  values  of  x,  y,  and  z  in  the  equations  {x-\-y 

-\-zf =S000,  y''-\-2yzS6  =  64>—z\  and  2a?y+202:=200. 

jins.  a?=10,  y=8,  and  z=2, 

4.  Find  the  values  of  x  and  y  in  the  equations  5a? — 5y=: 
^y,  and  a:^+4y^=:181.  .^ns.  x=9,  and  y-=5. 

5.  Find  the  values  of  a?,  y,  and  z  in  the  equations  j?'^y=54, 
y2;=8,  and  xz—1%  Ans.  x—Z,  y=2,  and  z—^. 

6.  Find  the  values  of  x  and  y  in  the  equations  a?^4-y^= 

,  and  xy= Ans,  07=3  or  —2,  and  y=2  or  — 3 

x—y  x—y 

x—y 

6 

xy= ; 

0? — V 


SECT.  VII.]  PURE    EQUATIONS.  207 

Multiplying  the  second  equa- >  o*  —   ^^    .  n^ 

tion  by  2  -         -         -)  ^~x^ '  ^  '^ 

Subtracting  the  third  from  )  <i  a  i  3  1  />«  \ 
the  first  -        -        -         -  >                         X — y 

Contracting  the  first  member      -  (x — y)*= ;       (5.) 

X — y 

Clearing  of  fractions   -        -        -  (x — yy=lj             (6.) 

Evolving X — y=lj             (7.) 

Substituting  for  x — y  its  val-  >  a:* -+-«»— 13  •           ^8  \ 

ue  in  the  first  equation     -  i 

Substituting  for  x—y  its  val-  >      ^  Oj.y_i2  ;           (9.) 

ue  in  the  third  equation   -  > 
Adding  the  eighth  and  ninth  >  ^^cixy-^f=25  ; 

equations         -         -         -  > 

Evolving x-\-y=±5. 

But x^y=l. 

Hence  -        •         -x=3or — 2,  and  y=2  or — 3. 

7.  Find  the  values  of  x  and  y  in  the  equations  x'^-{-y^= 
13,  and  a?^+y^=5.  ^ns.  a?=27  or  8,  and  y  =  8  or  27. 

8.  Find  the  values  of  x  and  y  in  the  equations  3c^-\-sl*y*-{- 
y»=273,  and  x'+xy+y'=21. 

^ns.x=±2  or  ±2v/— l,and  y=±l  or  ±  >/~l. 

9.  Find  the  values  of  x  and  y  in  the  equations  (x' — y*). 
(a?— y)  =  3xy,  and  (a?*— y«).(x-— y2)=45a:*y'. 

^ns.  x=z4f  or  2,  and  y=2  or  4. 

10.  Find  the  values  of  x  and  y  in  the  equations  3^y-\-xy'^= 
6,  and  a^y*+a:'y'=12.  Ans.  x=2  or  1,  and  y=l  or  2. 

11.  Find  the  values  of  a  and  y  in  the  equations  Vx — Vy 
=  3,  and  Vx-\-Vy=^7.  Am.  x=625,  and  y=16. 

r?         5   3 

12.  Find  the  values  of  x  and  y  in  the  equations  x*+a:*y* 

4-y'=1009,  and  r'-ha?V4-y*=  582193. 

^n«.  ar=8l  ar  16,  and  y=16  or  81. 


208  ELEMENTS    OF   ALGEBRA.  [sECT.  VII. 

PROBLEMS  PRODUCING  PURE  EaUATIONS. 
321. — 1.  What  two  numbers  are  those  whose  difference  is 
to  the  greater  as  2  to  9,  and  the  difference  of  whose  squares 
is  1281  ^^ns.  18  and  14. 

2.  A  fisherman  bfeing  asked  how  many  fish  he  had  caught, 
replied,  "If  you  add  14  to  the  number,  the  square  root  of 
the  sum,  diminished  by  8,  will  equal  nothing."  How  many 
had  he  caught  %  Ans.  50. 

3.  A  merchant  gains  in  trade  a  sum,  to  which  $320  bears 
the  same  proportion  as  five  times  the  sum  does  to  $2500. 
What  is  the  sum  \  Ans.  $400. 

4.  What  number  is  that,  the  fourth  part  of  whose  square 
being  subtracted  from  8,  leaves  a  remainder  equal  to  4  \ 

Ans.  4, 

5.  It  is  required  to  divide  the  number  18  into  two  such 
parts,  that  the  squares  of  these  parts  may  be  in  the  propor^ 
tion  of  25  to  16.  Ans.  10  and  8. 

6.  It  is  required  to  divide  the  number  14  into  two  such 
parts,  that  the  quotient  of  the  greater  part,  divided  by  the 
less,  may  be  to  the  quotient  of  the  less,  divided  by  the  great- 
er, as  16  to  9.  Ans.  8  and  6. 

7.  Two  persons,  A  and  B,  lay  out  some  rnoney  on  specu- 
lation. A  disposes  of  his  bargain  for  $11,  and  gains  as 
much  per  cent,  as  B  lays  out;  B  gains  $36,  and  it  appears 
that  A  gains  four  times  as  much  per  cent,  as  B.  Required 
the  capital  of  each.  Ans.  A's  $5,  and  B's  $120. 

8.  A  gentleman  bought  two  pieces  of  silk,  which  together 
measured  36  yards.  Each  of  them  cost  as  many  shillings 
by  the  yard  as  there  were  yards  in  the  piece,  and  their 
whole  prices  were  as  4  to  1.  What  were  the  lengths  of  the 
pieces  \  Ans.  24  and  12  yards. 

9.  A  number  of  boys  set 'out  to  rob  an  orchard,  each  hav- 
ing as  many  bags  as  there  were  boys  in  all,  and  each  bag 
capable  of  containing  as  many  apples  as  there  were  boys. 
They  filled  their  bags,  and  found  the  whole  number  of  ap- 
ples was  1000.     Wto  was  the  number  of  boys  \     Ans.  10., 


SECT.   VII.]  PURE    EQUATIO^JS.  209 

10.  Several  gentlemen  made  an  excursion,  each  taking 
the  same  sum  of  money.  Each  had  as  many  servants  as 
there  were  gentlemen  ;  the  number  of  dollars  which  each 
had  was  double  the  number  of  all  the  servants;  and  the 
whole  sum  of  money  taken  out  was  {^14'58.  What  was  the 
number  of  gentlemen  1  *dfu.  9 

11.  There  is  a  rectangular  field,  whose  length  is  to  the 
breadth  as  6  to  5.  After  planting  one  sixth  of  the  whole, 
there  remained  625  square  yards.  What  are  the  dimen- 
sions of  the  field  1 

*dns.  The  sides  are  30  and  25  yards. 

12.  There  are  two  numbers,  which  are  to  each  other  as  3 
to  2,  and  the  difference  of  their  fourth  powers  is  to  the 
sum  of  their  cubes  as  2G  to  7,     What  are  the  numbers  % 

An&.  6  and  4-. 

13.  What  two  numbers  are  as  5  ta  4,  and  the  sum  of 
whose  cubes  is  5103  \  Jlns,  15  and  12. 

14.  There  is  a  rectangular  field  containing  360  square 
rods,  and  whose  length  is  to  its  breadth  as  8  to  5.  What  is 
the  length  and  breadth.  Ans,  Length  24,  breadth  15. 

15.  There  are  two  square  fields,  the  larger  of  which  con* 
tains  13941  square  rods  more  than  the  smaller,  and  the  pro- 
portion of  their  sides  is  as  15  to  8.  What  is  the  length  of 
the  sides  *{  An$. 

16.  Two  travellers,  A  and  B,  set  out  to  meet  each  other. 
They  started  at  the  same  time,  and  travelled  on  the  direct 
road  between  the  two  places  \  and  on  meeting,  it  appeared 
that  A  had  travelled  18  miles  more  than  B,and  that  A  could 
have  gone  B's  distance  in  15^  days,  while  B  would  have 
been  28  days  in  going  A'^s  distance.  What  was  the  distance 
travelled  by  each  1  Jlns.  A*s  72,  B's  54. 

17.  There  are  two  men  whose  ages  are  to  each  other  as 
5  to  4,  and  the  sum  of  the  third  power  of  their  ages  is 
137781.     What  are  their  ages  1         Am.  45  and  36  years. 

18.  Find  two  numbers,  such  that  the  second  power  o(  the 
greater,  multiplied  by  the  less,  may  be  equal  to  448  j  and 

Dd 


210  ELEMENTS    OF    ALGEBRA.  [nECT.  VII. 

the  second  power  of  the  less,  multiplied  by  the  greater,  may- 
be 392. 

19.  A  man  wishes  to  make  a  cellar  that  shall  contain 
Sll04f  cubic  feet,  and  in  such  a  form  that  the  breadth  shall 
be  twice  the  depth,  and  the  length  li  the  breadth.  What 
must  be  the  length,  breadth,  and  depth  1 

jins.  Length  48,  breadth  36,  depth  18. 

20.  A  man  wishes  to  make  a  cistern  that  shall  contain 
500  gallons  of  wine,  in  such  a  form  that  the  length  shall  be 
to  the  breadth  as  5  to  4,  and  the  depth  to  the  length  as  2  to 
5.  Now,  allowing  231  cubic  inches  for  one  wine  gallon, 
what  will  be  the  length,  breadth,  and  depth  1 

AFFECTED  EaUATIONS  OF  THE  SECOND  DEGREE. 
322.  Let  2p  nnd  q  be  two  variable  numbers,  2/?  represent- 
ing the  coefficient  of  the  unknown  quantity,  and  q  the  known 
quantity ;  then,  however  complicated  may  be  the  equations 
which  involve  the  first  and  second  powers  of  the  unknown 
quantity,  they  may  be  reduced  to  one  of  the  four  following 
forms  : 

(1.)     x'-\-2px=q.  (3.)     x^-\-2px=-q. 


(2.)     x''  —  2px=q. 


(4>.)     x^  —  2px=z—q. 


Let  us  then  determine  the  process  by  which  equations 
of  these  forms  may  be  solved. 

323.  We  have  already  seen  that  a  binomial  cannot  be  a 
perfect  square,  and  also  that  the  root  of  a  trinomial,  which 
is  a  perfect  square,  may  be  formed  by  taking  the  root  of  the 
two  terms  that  are  complete  powers,  and  connecting  them 
by  the  sign  of  the  other  term  (Art.  292).  Thus,  \/x^-\-2px 
-{-p^=x-\-p,  and  \/x^—2px-{-p^=:x—p. 

324.  We  have  also  seen  that  the  square  of  a  binomial  is 
equal  to  the  square  of  the  first  term,  plus  twice  the  product 
of  the  two  terms,  plus  the  square  of  the  last  term.     Thus, 

(x-{-pf—x^-\-2px-{-p^ ; 
And  the  square  of  the  residual,  x—p^  gives 
(x-py=x^—2px+p\ 


SECT.  VII.]    AFFECTED    EQUATfONS    OF    SECOND    DEGREE.    211 

Hence,  if/?'  be  added  to  both  members  of  each  of  the  pre- 
ceding four  forms  of  the  affected  quadratic  equation,  the 
first  member  of  each  will  be  a  perfect  square.     Thus, 

(5.)     :3^+2px+f=q-{.f', 

(6.)     x«— 2pa:+;>*=9-hp'; 

(7.)     s^+lpx^p'^f—q; 

(8.)     j?-^px-\-p^=f^^, 

325.  If  we  compare  p^  with  the  coefficient  of  x,  it  will  be 
found  equal  to  the  square  of  half  of  it.     Thus, p^=  IJl\, 

Hence,  when  the  quadratic  equation  is  reduced  to  the  first, 
second,  third,  or  fourth  power,  the  first  member  may  be 
rendered  a  perfect  square  by  adding  the  square  of  half  the 
coefficient  of  the  first  power  of  the  unknown  quantity  to 
both  members  of  the  equation.  This  is  called  completing 
the  square, 

326.  Each  of  the  above  equations  may  be  reduced  by  ex- 
tracting the  square  root  of  both  members,  and  making  the 
necessary  transformations. 

Extracting  the  square  root  of  the  (5),      x-\-p=  ±  Vq-{-p^ ; 
Transposing         -         .         .         .         .       x=z—p±y/q-^p^. 
Extracting  th6  square  root  of  the  (6),      x—p=  ±  y/q-\-p* ; 

Transposing x=;)±  v/^-f-p\ 

Extracting  the  square  root  of  the  (7),      x-\-p=  ±  y/p^ — q  ; 


Transposing x=—p±>/p^—q- 

Extracting  the  square  root  of  the  (8),     x — p=  ±  ^/p^—q  ; 
Transposing x=p±  y/p^ — q. 

327.  Hence,  for  the  solution  of  affected  quadratic  equa- 
tions, we  have  the  following  general 

RULE. 

1.  Reduce  the  equation  to  one  of  the  above  four  forms. 

2.  Complete  the  square  by  adding  to  both  members  of  the  equa- 
tion the  square  of  half  the  coefficient  of  the  first  power  of  the 
unknown  quantity. 


212  ELEMENTS    OF    ALGEBRA.  [sECT.  VII. 

3.  Extract  the  square  root  of  both  members^  observing  to  affect 
the  second  member  with  the  double  sign  ± ,  and  complete  the  re-' 
duction  by  the  preceding  principles. 

JVote. — Equations  of  this  nature  also  give  two  values  of 
the  unknown  quantity.     Thus,  x^-\-4<x=  12. 

Completing  the  square  -  a?^+4!a?+4=    12+4  =  16. 

Extracting  the  square  root      -  x-\-2=±  \/16  =  d=4. 

Transposing-         -         -         -         x— — 2±4=i2  or — 6. 

EXAMPLES. 

1.  Find  the  values  of  x  in  the  equation  3a?^-|-18a;=;81. 

3a?2+18a:=81 
x^+  ex=21 
a?'+6a?+9  =  27+9  =3G 
x-\-3=±VS6=:±6 
x——3±6=:z3or—9. 

2.  Find  the  values  of  x  in  the  equation  3a?^+2a; — 9=76. 

^ns.  x—5y  or  — 5|. 

Of: 

3.  Find  the  values  of  x  in  the  equation  -= — 4. 

^  2     x-\-2 

Ans.  07=4,  or  — 14. 

4.  Find  the  values  of  x  in  the  equation  a?^-f-48=426  +  12a? 
— bx^.  Ans.  x=:^,  or  — 7. 

5.  Find  the  values  of  x  in  the  equation  2a:^+12a7= — 16. 

Ans.  X— — 2,  or  — 4. 

6.  Find  the  values  of  a?  in  the  equation  x^ — 150?= — 54. 

Ans.  07  =  9,  or  6. 

2?^       X       X^         V         IS 

7.  Find  the  values  of  a?  in  th«  equation  _.+_:=_ — 1_+ — 

2     4     5     10     20 

Ans.  x=  1,  or  — 2-}. 

X         X  3(? 

8.  Find  the  values  of  a?  in  the  equation  — +- — 15=— 4-^ 

2     3  4 

— 14f.  Ans.  a?=3,  or  — i. 

9.  Find  the  values  of  x  in  the  equation  4a^ — 2a?^+2aa?= 
l^ab—l%W.  Ans.  x=2a—3b,  or  —a-\-3b. 

10.  Find  the  values  o(  x  in  the  equation  2ax — x^=z — 2ab— 
h^,  Ans.  2a-\-byOx  — h. 


SECT.  VII.]     AFFECTED  EQUATIONS  OF  SECOND  DEGREE.      213 

A  SECOND  METHOD  OF  COMPLETING  THE  SaUARE. 

328.  It  is  frequently  impossible  to  clear  the  highest  power 
of  the  unknown  quantity  from  its  coefficient  without  intro- 
ducing fractional  expressions  into  the  equation.  But,  resu- 
ming the  four  forms  of  affected  quadratic  equations,  and  let- 
ting a  represent  the  coefficient  of  «*,  we  shall  have 


(1.)     ax'+2px=q; 
(2.)     ax'—2px=q; 


(3.)     aa^+2px=^q; 
(4.)     ax* — 2px= — q. 


Multiplying  each  of  these  equations  by  4a,  and  adding  the 
square  of  2p  to  both  members,  we  shall  have 
(5.)     4>a^x^-{-8apx-\-4fp'^=4>aq-\-4>p' ] 
(6.)     4fa^a^—Sapx-\-4>p*=4>aq+4<p'; 
•^(7.)     4aV-f8a;>a?+4>'=— 4^0^+ V ; 
(8.)     4a-x* — S(ipx-\-4tp'^  =  — 4faq-\-4<p^. 
329.  It  is  evident  that  the  first  member  of  each  of  these 
equations  is  a  perfect  square  ;  hence,  extracting  the  square 
root  of  both  members,  we  shall  have 

(9.)     2ax+2;>=  ±  v/4a9-f-4/ ; 
(10.)     2ax—2p=  ±  v/4a9-f  4/  ; 


(11.)     2ax-\-2p=  ±  V  —4faq-\-4p* 
(12.)     2ax—2p=±  V—^aq-\-4^, 
Or,  reducing, 


(13.) 
(14.) 
(15.) 


,_ — 2p±  y/^aq-\-^p^  ^ 
2a ' 


_+2p±>/^aq-\-^p\ 
2a 


— 2p±  y/—4>aq-\-^p* 


2a 


(16)  ^_-f2;)dbv/— 4a^+V 

2a  ' 

330.  Hence  the  square  of  an  affected  quadratic  equation 
may  be  completed  by  the  following  general 

RULE. 

1.  Multiply  both  members  of  the  equation  by  four  times  the 


214  ELEMENTS    OF    ALGEBRA.  [sECT,  VII. 

coefficient  of  the  second  power  of  the  unknown  quantity^  and  add 
the  square  of  the  coefficient  of  the  first  power  to  loth  members  ; 
the  first  member  will  then  be  a  perfect  square. 
2.  Extract  the  root^  and  reduce  as  before. 

EXAMPLES. 

1.  Find  the  values  of  a?  in  the  equation  2a;^+3ir=65. 

2a?2+3a:z=65. 

Completing  the  >      16^2+24^+9^520  +  9   =529; 

square      -        > 
Evolving     -         -        -         4a?+3=±v/'529=±23,- 

4a?=— 3±23=20,or— 26; 
x=b^  or  — 6|. 

2.  Find  the  values  of  a:  in  the  equation  3a?^ — ^x — 4—80. 

Ans.  a?=7,  or  — 4. 
35 ^ 

3.  Find  the  values  of  a;  in  the  equations  4a: — =46. 

X 

Ans.  x—\%  or  — f. 

2ii?     X 
4f.  Find  the  values  of  x  in  the  equation  x^-\-— — .^=8+ 

12       6      *  J  11 

Sep 

5.  Find  the  values  of  x  in  the  equation  2a?^+8a?+7z= — 

4 

—^+197.  Ans.  a;=8,  or  — lly^. 

8 

6.  Find  the  values  of  x  in  the  equation  - — _+7-|=8. 


Ans.  a?=:l^,  or  — |. 
c 
7.  Find  the  values  of  x  in  the  equation 


8— a;    2a?— 11 


2  a?— 3 

^       .  Ans.  a;=6,  or  \. 

6 

8.  Find  the  values  of  x  in  the  equation  5a?-+4a?=273. 

Ans.  x—1,  or  — 7|. 

PARTICULAR  CASES  OF  AFFECTED  aUADRATICS. 
331.  It  is  evident  that  every  equation  of  the  form 

a?'"+2^a?"=9 


r 


SECT.   VII.]  AFFECTED   EQUATIONS.  215 

may  be  solved  by  the  preceding  rules  ;  for,  let  y=x",  and 

y^=x^j  and  the  above  equation  will  become 

f-\-2py=q; 
Completing  the  square,  i/^-{-2py-^p^=:q-^p'^ ', 
Evolving  and  transposing       -        y= — p±:  \/g-\-p*i 
Substituting  the  value  of  y    -       x'*= — p±Vq-j-p'; 


Evolving        -         -        -         .        x=\/^ — p±Vq+p'' 

332.  Equations  also  occur  in  the  form 
>  I 

ar-\-2px^=q. 

1  s 

Let  y=ir",  then  y'=af* ;  and  substituting  these  values, 

y'-\-2py=q;  

Reducing         -        -        -        -    y=—p±Vq-\-p^, 


Or ar=-p±Vq-\-p^j 

Involving  to  the  nth.  power      -     xz=(—p±y/q-\-p^y. 
These  equations  may  be  readily  solved  without  the  for- 
mality of  substitution.     Resume  the  equation 

a^-\-2px''=q  ; 
Completing  the  square,  af''-\-2px^-{-p^=q-{-p'^ ; 
Evolving        -         -         -         .  af'-\-p=±'^q-\-p^ ' 


Transposing-        -        -        -        JL'^=—p±\/q-\-p'; 

Evolving        -        -        -        -         x=  \/  —p  ±  ^q-\-f* 

Resume,  also,  the  equation 

s  t 

a^-\-2po^z^q  ; 

3  1 

Completing  the  square,  tj^-^2px''-\-p*=.q-\-p^  \ 

1  

Evolving        -         -         -         -  a?^-(-^=dr\/^ -!-/>'; 


Transposing-         -         -         -        x'^zzL—p^s/q-^-p^'^ 
Involving       -         -         -         -  x—{^—p^s/q-\-p^y, 

333.  The  same  principles  will  apply  also  to  all  equations 
in  which  there  are  two  terms,  simple  or  compound,  and  the 


216  ELEMENTS    OF    ALGEBRA.  [sECT.  VII. 

exponent  of  one  is  double  that  of  the  other.     Thus,  in  the 
equation 

Letting    yz=zx'^-{-2px-\-q^  and  f=^{x^-\-'Zpx-\-qy,  we   shall 

have  then    -         -        y^-{-y=:q', 
And         .         -         -         -     'yz=-^±^//-fiJ 
Whence  -        ^-^';ipx-\-q=-^±Vq'+}, 


And         ...         -      x=-p±\/p^q-i±Vq'+ly 
Or,  in  the  equation  'W 

(ax  +  2by—2p(ax-{-2b)  =  q, 
Letting  y='ax-\-2b^  and  y'^=(ax-^2bf^  we  shall  have 

y^-2py=^q, 
And  -  -  .  .       y—p±y/q-^p2- 

Whence  -        -      ax-{-2b=p±Vq+p'^, 

And        ...        -     x=P±^l±Pl:i^. 

a 

These  equations  may  also  be  solved  without  the  formality 
of  substitution. 

334.  If  the  indeterminate  quantity  y=0,  the  affected  qua- 
dratic will  assume  the  form 

x''±2px=0. 
This  equation  may  be  readily  solved ;  for,  dividing  both 
members  by  x^  we  have 

x±2p  =  0  j 
Transposing        -         -         x—:p2p. 

335.  Equations  involving  more  than  one  unknown  quan- 
tity, as  xY-\-^pxy=q,  or  (a?"+y")^+2p(a?"4-y")=5',  if  the  ex- 
ponent of  one  term  is  double  that  of  the  other,  may  also  be 
reduced  to  simpler  forms  by  completing  the  square  and  per- 
forming the  necessary  transformations. 

336.  When  there  are  two  unknown  quantities  similarly 
involved  in  the  equations,  the  work  may  be  simplified  by 
the  introduction  of  two  additional  symbols  which  shall  rep- 


SECT.  VII.]  AFFECTED    EQUATIONS.  217 

resent  known  functions  of  the  unknown  quantities.     Thus, 
in  the  equations 

Let  x-{-y=2s,  and  x — y=2z  ;  then  x=s+z,  and  y=s — z. 
And       -  x'=(s-{-zy=s'-^2sz-^z' ; 
And       -  y^z=:(s—zy=s^—2sz  +  2^'y 
Adding  -         -  x'-\-f=2s^-\-2z'  j 
Consequently,  bz=2s^-{-22^  j 

Transposing  and  >  ^^^2^^  ^^^  z=±./^^; 
reducing     -      j  2  V       2 

Hence    -        -         j=^+-v/-Z_,  and  y=s—y/  ~~      ; 


^                  ,           /b—4>,  or  ]a^       j        ,           /^— 4-,  or  ia* 
Or  T=ia-y^ V"^'  and  y=ia— y/ V~^* 

Similar  operations  will  reduce  any  equations  of  the  same 
form ; 

As,  x  +  y=a,  and  a:'  +  y*=J;  or  x-\-y=ay  and  x*-{-y*=b  ;  or 

X'{-y=ay  and  j^-f  y^=Z>,  &c. 
337.  Again,  let  us  take  the  equations 

X  -\-y  =ai 

y    ^ 

Clearing  the  second  equa-  >    ,      ,_, 

tion  of  fractions       -        >         y  —    y' 
Let  x-\-y=2sy  and  x — y=2z,  as  before;  then  x=s  +  z,  and 

y=s—z. 
Whence        -         a^={8+zY=s^-^2sz-\-:^; 
And      -         -        f=(s—zYz=zs''—2sz-\-z'. 
Adding,  x'-\-f=2s^-\-22^=b(s-\-z).(8—z)z=b{s'^2^)  = 

bs'-bs^.  

Whence        -        2r= __  :  or  2:=i  *  7^-; — —J 

2-\-b     '  V     *+2 

Or,  substituting  and  reducing, 

,=,±  v/ip2,  and  y=,T  ,/]pE. 
V     *+2  *       ^V     i+2 

19  Ee 


218  ELEMENTS    OF    ALGEBRA.  [SECT.  VII. 

Restoring  the  value  of  s  and  5^, 

Similar  operations  will  reduce  any  equations  of  the  same 
form, 

As,  x-\-y—a^  and  —+^=5,  &c. 
y     X 
There  are  a  variety  of  expedients  by  which  complicate 
equations  may  be  simplified.     The  above  cases  will  indicate 
some  of  the  most  general.     Others  must  be  left  to  exercise 
the  skill  and  ingenuity  of  the  learner. 

AFFECTED  EaUATIONS  INVOLVING  ONLY  ONE  UN- 
KNOWN aUANTITY. 

35 3^ 

1.  Find  the  values  of  x  in  the  equation  6a7+ =4)4<. 

X 

Clearing  of  fractions         -         -         6a?^4-35 — 3a?=44a?; 

Transposing Qx^ — 47a?= — 35  ; 

Completing  the  square  and  reducing         -         x=l,  or  |. 

3^, 3 

2.  Find  the  values  of  x  in  the  equation  5a? — =2a?4- 

X — 3 

^^—^,  Ans,  a?z=4,  or  —1. 

2 

3j,      2  2a? 2 

3.  Find  the  values  of  x  in  the  equation  — +__=a?4- — _ — . 

2      da?  3 

Ans.  a?n:2±2v/2. 

3^ 10 

4.  Find  the  values  of  a?  in  the  equation  3a? — _ — ^^_  =  2-f- 

\) — ZiX 

5^i5.  ^;i5.  a?=lli,  or4.. 

2a?— 1 

5.  Find  the  values  of  a?  in  the  equation  a?^ — 4a:'' ==621. 

Ans.  a?n:3,  or  ^—23. 

6.  Find  the  values  of  x  in  the  equation     —=—__. 

2     4  32 

Ans.  x=:^~\^^'\^^^ll. 

2.  i 

7.  Find  the  values  of  a?  in  the  equation  2a?3  +  3a?3z=2. 

Ans.  a;=|,  or  — 8. 


SECT.  VII.]  AFFECTED   EQUATIONS.  219' 

8.  Find  the  values  of  x  in  the  equation  a:*-|-a:*=756. 

Jlns.  j:=243,  or  (—28)^ 

9.  Find  the  values  of  x  in  the  equation  (lO-hx)' — (10-f-x)* 
=2.  jJns.  x=6. 

10.  Find  the  values  of  x  in  the  equation  2(1 -fa? — a?*) — 

(l+i— x»)^=— |.  J^ns.  x=i-\-lV'U, 

11.  Find  the  values  of  x  in  the  equation  v^ar* — a*=a? — b. 

Jins.  x=-±    / . 

2      /      126 

12.  Find  the  values  of  ar  in  the  equation  2Vx—a-\-3y/2x 

7a-f-5x 
= — =.  *d7is,  x=9a. 

y/x — a 

Ajr 5       2x 7 

13.  Find  the  values  of  x  in  the  equation  —  i 1= 

X         3a:+7 

?^-.  jJns.  x=2. 

13x 

q  /I 

14.  Find  the  values  of  a?  in  the  equation -+— = 

—,  '  J3ns.  x=z3. 

5x 

AFFECTED  EaUATIONS  INVOLVING  TWO  OR  MORE 
UNKNOWN  aUANTITlES. 


1.  Given 

And  - 


^t^^^  +  2^r   ''Jtofindxandy. 
x'-f.    xy  +  f=133S 


Dividintr  the  second  by  the  first  /            , —  .  _  ._  . 

°                                          }    x—Vxy-\-y=  7;  (3.) 

equation         -         -         .         -^          ^    ^^^  »  ^    ' 

Adding  the  first  and  third  equa-  /          Oj:4-2v=  26  (4>  ) 

tions \                    ^  >  V  •; 

Or x-{-y=  13;  (5.) 

Substituting,  in  the  first  equation,         Vary  4-13=  19;  (6.) 

Whence ^1^=  6,  (7.) 

And xy=  36;  (8.) 

Multiplying  by  3       -        -         -         -           3a?y=108;  (9.) 

Subtractinsf  the  ninth  from  the  /     «     «          a  ok  /ia\ 

,^.                            >  a^'-'Xxy^f=  25;  (10.) 
second  equation     -         -         •\ 


220  ELEMENTS    OF    ALGEBRA.  [sECT.  VII. 

Evolving  ------        X — y=d=5.      (11.) 

Adding  the   eleventh   and  fifth;    2a:::.13±5  =  18,  or  8; 

equations       -         -         -         -\ 

Whence a?=9,  or  4. 

Subtracting  the    eleventh  from;    2^=13^5  =8,  or  18; 

the  fifth  equation  -         -*        -  ) 
Whence 2/=^>  ^^  ^- 

2.  Given        -        x^^x^y^  18-/  >  ^^^^^  ^  ^^^  ^_ 
And  -         -         -  xy—   6         \ 

Transposing  in  the  first;      ^2^3,2^^^^^  jg  .  (3.) 

equation  -         -    ) 

Multiplying  the  second)    _        _        2a^y.=  12;  (4.) 

equation  by  2         -    ) 
Adding   third  and;   ^j^^^j^fj^^j^y^^^^  (5.) 

fourth  equations  \ 
Or-         -         -        -       (a?  +  2/)^+(a7+2/)=30;  (6.) 

Completing  the      /  (^^y)2^(^+2^)  +  i  =  3o+i=.A^5     (7.) 
square  -       )  ^ 

Evolving  -         -         -  a7+2/-hi=±Y'  ^'^ 

And       -        ^+y=±n_|=.:^or-^=5,or-6;         (9.) 

Whence,  from  the  first  }  x^^f=\%  or  24  ;      (10.) 

equation  -  -  ) 
Subtracting  the  4th  from  )      ^._2^^_^y2^i^  ^^  12  ;        (11.) 

the  10th  equation  ) 

Evolving  -        ^— y  =  ±l,  or±x/12=:±273;  (12.) 

Adding    13th    and  ;  2a^=5±l,  or  — 6±2v/3  ; 

9th  equations        > 
Whence  -        -         -     a^=3  or  2,  or  — 3±  n/3  ; 

Subtract'g  13thfrom)  2y^4  or  6,  or  — 6:f  2>/3  ; 

9th  equation  \ 

Whence  -        -        -     2/=^  ^^  ^'  ^^  — 3:F  V3. 

3.  Given      -         ■     ^xy^m-x^f  \  ^^  ^^^  ^  ^^^  ^^ 
And  -         -    x-\-y—  6  \  ^^ 

Ans,  a:=4  or  2,  or  3±v/21  j  and  2(=2  or  4,  or  3=f\/21. 


SECT.  VII.] 

4.  Given 
Aud 

5.  Given 
And 

6.  Given 
And 


AFFECTED   EQUATIONS. 


22! 


7.  Given 
And 


>  to  find  X  and  y. 


x-fy  =     S 

jJns.  x=5  or  3,  and  y=3  or  5. 

..""«.,  >  to  find  X  and  y. 

Jlns,  x—b  or  2,  and  y=2  or  5. 

c       .""-^-o  >  to  find  X  and  v. 
x'+y'=1056\  ^ 

*^ns.  x=^  or  2,  and  y=2  or  4. 


xV  = 


=2y^j 


to  find  X  and 


8x3- 
^ns.  x=2744.  or  8,  and  y=4  or  9604. 

PROBLEMS  PRODUCING  AFFECTED  EaUATIONS. 

1.  It  is  required  to  divide  the  number  40  into  two  such 
parts  that  the  sum  of  their  squares  shall  be  818. 

Jlns.  23  and  17. 

2.  What  two  numbers  are  those  whose  difference  is  9, 
and  their  sum,  multiplied  by  the  greater,  produces  266  1 

Ans.  14  and  5. 

3.  An  officer  would  arrange  1200  men  in  a  solid  body,  so 
that  each  rank  may  exceed  each  file  by  59  men.  How  many 
must  be  placed  in  rank,  and  how  many  in  filel 

^ns.  Rank  75,  file  16  men. 

4.  Some  bees  had  alighted  upon  a  tree  ;  at  one  flight  the 
square  root  of  half  of  them  went  away;  at  another  eight 
ninths  of  them  ;  two  bees  then  remained.  How  many  alight- 
ed on  the  tree  1  ^ns.  72. 

5.  A  mercer  bought  a  piece  of  silk  for  j£16  4*.,  and  the 
number  of  shillings  he  paid  per  yard  was  to  the  number  of 
yards  as  4  to  9.  How  many  yards  did  he  buy,  and  what 
was  the  price  per  yard  1       »dns.  27  yards,  at  12*  per  yard. 

6.  There  is  a  field  in  the  form  of  a  rectangular  parallelo- 
gram, whose  length  exceeds  the  breadth  by  16  yard?,  and  it 
contains  960  square  yards.    Required  the  length  and  breadth. 

^ns.  Length  40,  breadth  24  yards. 

7.  A  person  being  asked  his  age,  answered,  "  If  you  add 


222  ELEMENTS    OF    ALGEBRA.  [sECT.  VII. 

the  square  root  of  it  to  half  of  it,  and  subtract  12  from  the 
sum,  there  will  remain  nothing."     What  was  his  age  1 

^ns.  16. 

8.  What  number  is  that  which,  if  divided  by  the  product 
of  its  digits,  the  quotient  will  be  2;  but,  if  27  be  added  to 
the  number,  the  digits  will  be  inverted  "i  Ans.  36. 

9.  Find  two  numbers  such  that  their  sum,  their  product, 
and  the  difference  of  their  squares  may  all  be  equal  to  one 
another.  ^?J5.  ^±|v/5,  and  |±^-v/5. 

10.  A  and  B  hired  a  pasture,  into  which  A  put  4  horses, 
and  B  as  many  as  cost  him  I85.  a  week.  Afterward  B  put 
in  two  additional  horses,  and  found  that  he  must  pay  20*. 
a  week.  How  many  horses  had  B  at  first,  and  at  what 
rate  was  the  pasture  hired  \ 

Ans.  B  had  6  horses,  and  the  pasture  was  hired  at 

5O5.  per  week. 

11.  A  labourer  dug  two  trenches,  one  of  which  was  6 
yards  longer  than  the  other,  for  £>Yi  I65.,  and  the  digging  of 
each  of  them  cost  as  many  shillings  per  yard  as  there  were 
yards  in  its  length.     What  was  the  length  of  each  \ 

Ans,  10  and  16  yards. 

12.  A  and  B  set  out  from  two  towns  which  were  distant 
from  each  other  247  miles,  and  travelled  the  direct  road  till 
they  met.  A  went  9  miles  a  day,  and  the  number  of  days 
at  the  end  of  which  they  met  was  greater  by  3  than  the 
number  of  miles  which  B  went  in  a  day.  How  many  miles 
did  each  go  ]  Ans.  A  117,  and  B  130  miles. 

13.  Two  merchants  each  sold  the  same  kind  of  stuff;  the 
second  sold  3  yards  more  of  it  than  the  first,  and  together 
they  receive  35  crowns.  The  first  said  to  the  second,  "  I 
would  have  received  24  crowns  for  your  stuff;"  the  other 
replied,  "I  would  have  received  12g  crowns  for  yours." 
How  many  yards  did  each  of  them  sell  \ 

Ans.  The  first  sold  15  or  5,  the  second  18  or  8. 

14.  A  widow  possessed  $13,000,  which  she  divided  into 
two  parts,  and  placed  them  at  interest  in  such  a  manner  that 


SECT.  VII.]  AFFECTED    EQUATIONS.  223 

the  incomes  from  them  were  equal.  If  she  had  put  out  the* 
first  portion  at  the  same  rate  as  the  second,  she  would  have 
drawn  for  this  part  $360  interest ;  and  if  she  had  placed  the 
second  out  at  the  same  rate  as  the  first,  she  would  have 
drawn  $490.     What  were  the  two  rates  of  interest  I 

^ns.  7  and  6  per  cent. 

15.  The  sum  of  two  numbers  is  9,  and  the  sum  of  their 
cubes  24-3.     What  are  the  numbers  1  j9ns.  3  and  6. 

16.  The  sum  of  two  numbers  is  10,  and  the  sum  of  their 
fourth  powers  is  1552.     What  are  the  numbers  1 

jJns,  4  and  6. 

17.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
fifth  powers  3157.     What  are  the  numbers  1 

^ns.  5  and  2. 

18.  There  are  two  square  buildings  that  are  paved  with 
stones  a  foot  square  each.  The  side  of  one  building  ex- 
ceeds that  of  the  other  by  12  feet,  and  both  their  pavements 
together  contain  2120  stones.  What  are  the  lengths  of  them 
separately  ]  J^ns,  26  and  38  feet. 

19.  A  regiment  of  soldiers,  consisting  of  1066,  formed 
into  two  squares,  one  of  which  has  four  men  more  in  a  side 
than  the  other.  What  number  of  men  are  in  a  side  of  each 
of  the  squares'?  j^ns.  21  and  25. 

20.  The  plate  of  a  looking-glass  is  18  inches  by  12,  and  is 
to  be  framed  with  a  frame  of  equal  width,  whose  area  is  to 
be  equal  to  that  of  the  glass.  Required  the  width  of  the 
frame.  Ans.  3  inches. 

21.  A  square  courtyard  has  a  rectangular  gravel-walk 
round  it.  The  side  of  the  court  wants  two  yards  of  being 
six  times  the  width  of  the  gravel  walk,  and  the  number  of 
square  yards  in  the  walk  exceeds  the  number  of  yards  in 
the  periphery  of  the  court  by  164.  Required  the  area  of 
the  court.  ^ns.  256  yards. 

22.  There  are  four  towns  in  the  order  of  the  letters  A,  B, 
C,  and  D.  The  difference  between  the  distances  from  A  to 
B  and  from  B  to  C  is  greater  by  four  miles  than  the  dis- 


224  ELEMENTS    OF    ALGEBRA.  [sECT.  VII. 

tance  from  B  to  D.  Also,  the  number  of  miles  between  B 
and  D  is  equal  to  two  thirds  of  the  number  between  A  and 
C ;  and  the  number  between  A  and  B  is  to  the  number  be- 
tween C  and  D  as  seven  times  the  number  between  A  and  C 
is  to  208.     Required  the  respective  distances. 

Ans.  A  B  42,  B  C  6,  C  D  26  miles. 

DISCUSSION  OF  THE  GENERAL  EaUATION  OF  THE 
SECOND  DEGREE.* 

CASE   I. 

338.  It  has  already  been  remarked,  and  we  will  now  pro- 
ceed to  demonstrate,  that  every  affected  equation  of  the  sec- 
ond degree  necessarily  admits  of  two  values  for  the  unknown 
quantity,  and  only  two. 

339.  Let  us  resume  the  first  of  the  four  forms  of  the  af- 
fected quadratic  (Art.  322). 

X'^-1J)X^q',  (1.) 

Adding  ;?^  to  both  members,      Q^-\-^])x-\-f-—q-\-f-^  (2.) 

Or   -         -         -        -     ■    -^        -         {x^^y^q^f-,  (3.) 

Let -         -        m^=q^f',  (4.) 

Then {x-^-ff^m^-,  (5.) 

Transposing     -         -         -  (ar+jp)^— ^'=0 ;  (6.) 

Resolving  into  factors,  (a? H-^+7w).(a?4-p — m)=:0  j  (7.) 
Dividing  by  a?+pH-?w        -         -      a>+^ — m  —  ^\ 
Transpasing     -         -         -  a?= — 'p-\-m^ox  x— — 'p-^  \fq-\-'p^  '^. 
Dividincr  the  7th  equation  /                   ,      .         ^ 
by  x-^-p — m  -         -          ) 


Transposing     -         -         -  x= — p — m,  or  a?= — j) — \/ q-{-p^. 
Either  of  these  values  of  x  will  answer  the  conditions  of 

the  equation. 

The  same  course  of  demonstration  might  be  applied  to 

the  remaining  three  forms  of  the  quadratic  equation. 

Hence  every  affected  equation  of  the  second  degree  necessarily 

admits  of  two  values  of  the  unknown  quantity^  and  only  two. 

CASE  II. 

340.  We  will  now  resume  the  results  obtained  in  the  four 
^  «  This  discussion  is  substantially  that  of  ]M.  Bourdon. 


SECT.  VII.]  AFFECTED    EQUATIONS.  225 

preceding  formulas,  and  enter  into  such  an  analysis  of  them 
with  reference  to  the  relative  values  of  q  and  p  as  will  de- 
termine the  particular  values  of  x.  These  results  are  (Art. 
326),  __ 

(1.)      x=-^±Vqj^',  I      (3.)      x=—p±V^ 


(2.)     x=+p±Vq-\-p'',         I     (4.)     X=+p±^p'-^. 

1.  Since  the  value  of  x  in  each  equation  is  expressed  by  a 
rational  term,  with  which  a  radical  is  connected  by  the  sign 
db,  in  order  that  this  value  may  be  found,  the  quantity  un- 
der the  radical  sign  must  be  positive. 

As  p^  is  necessarily  positive,  the  value  of  x  may  always 
be  found  in  the  first  and  second  equations. 

If  q-\-p^  is  a  perfect  square,  the  exact  value  of  x  will  be 
obtained  ;  if  it  is  a  ^urd,  its  approximate  value. 

In  the  third  and  fourth  equations,  if 

q<p\ 

the  value  of  x  may  also  be  found,  either  exactly  or  approxi- 
mately J  but  if 

q>p\ 

the  value  of  x  will  be  imaginary,  since  it  will  involve  the  ex- 
traction of  the  square  root  of  a  negative  quantity. 

2.  In  the  first  and  second  equations,  since 

P<y/g-{-p\ 
the  value  of  x  will  be  positive  when  the  radical  is  taken  pos- 
itive, and  negative  when  the  radical  is  taken  negative. 

3.  In  the  third  equation,  since 

pyy/p'—qy  « 

if  y<;>',  the  value  of  x  will  be  negative ;  but  if  g'^p^,  the 
value  of  X  will  be  imaginary. 

4.  In  the  fourth  equation,  since 

p>  y/p^—q, 
if  y<p^  the  value  of  x  will  be  positive ;   but  if  q'>f^  the 
value  of  a:  will  be  imaginary. 

5.  If  y=;>',  the  radical  expression  in  the  third  and  fourth 
equations  will  be  reduced  to  0,  and  the  values  of  x  will  be, 

Ff 


226  ELEMENTS    OF   ALGEBRA.  [sECT.  VII. 

In  the  third    -         -         -         -       xz= — p^ 
In  the  fourth  -         -         -         -       x=-\-p. 

6.  If  q—0,  the  equation  will  assume  the  form  x^±2px= 
±  0  ,•  and,  consequently,  x=  ±  2/?,  or  ±  0. 

7.  If  jo=:0,  the  equation  will  assume  the  form  x'^—ztq; 
and,  consequently,  x=±V±qi  and  the  value  of  x  will  be 
imaginary  in  the  third  and  fourth  forms  of  the  quadratic. 

8.  In  the  equation  ax'^±2px=±q,  if  a—0,  the  equation 
will  assume  the  form  ±2px—dzq,  or  be  reduced  to  a  simple 
equation. 

CASE    III. 

341.  In  order  to  show  why  we  obtain  the  imaginary  re- 
sults in  the  third  and  fourth  equations  when  q^p^,  we  will 
demonstrate  that  these  equations,  when  q^p^,  express  con- 
ditions that  are  incompatible  with  each  other. 

Kesume  the  equation 

X^ — 2pX=: q  ; 

Reducing        -         -         -         x=p±  y/p^ — q. 

Designating  the  first  value  of  x  by  x',  and  the  second  value 
by  x"j  we  shall  have 

x'=p+Vp^—q; 


x"=p — Vp  —q  S 


Adding        -         x'-{-x"=zp+  ^  f—qj^p—>j  f—q—^p. 
Hence  the  sum  of  the  two  values  of  x  is  equal  to  the  coeffi- 
cient of  the  first  power  of  the  unknown  quantity ,  taken  with  the 
contrary  sign. 

Multiplying  the  above  two  equations, 

Hence  the  product  of  the  two  values  of  x  is  equal  to  the  sec- 
ond member  of  the  equation,  taken  with  the  contrary  sign. 

Therefore,  in  the  general  equation,  x^ — 2px=: — q,  2p  is 
the  sum  of  two  numbers,  of  which  q  is  the  product.  Now 
it  has  already  been  demonstrated,  that  if  a  quantity  be  re- 
solved into  two  factors,  their  product  will  be  the  greatest 
possible  when  the  factors  are  equal  (Art.  206). 


SECT.  VII.]  AFFECTED    EQUATIONS.  227 

Hence  the  conditions  of  the  equation  limit  the  value  of  q  ; 
it  may  vary  between  the  limits  0  and  p*,  but  can  never  be- 
come greater  than  (-^J  =P^' 

If,  then,  we  assign  to  y  a  value  greater  than  the  square  of 
half  2p,  the  equation  will  express  conditions  which  are  in- 
compatible with  each  other,  and,  consequently,  the  value  of 
X  will  be  imaginary  or  impossible.     Thus, 

Let  it  be  required  to  divide  16  into  two  such  parts  that 
their  product  shall  be  72. 

Let        -         -        x=  one  of  the  parts, 
Then      -  16— a?=  the  other; 

And,  by  the  conditions  of  the  problem, 

x.(16— x)z=72; 

Multiplying       ...       16a:— a?^=72; 

Changing  the  signs  -         -        o^ — 16x= — 72  j 

Completing  the  square,     a^ — 16x+64=64 — 72= — 8; 

Evolving X — 8=db\/ — 8; 

Transposing      -        -        -        -         a7=8±  ■/— 8. 

Thus  we  obtain  an  imaginary  result,  which  should  be  the 
case,  as  16  can  be  divided  into  no  two  factors  whose  prod- 
uct shall  be  equal  to  72;  for,  since  2/)=16, ;)  will  equal  8, 
and  /?*:=64,  which  is  the  greatest  possible  product  that  can 
be  formed  of  two  numbers  whose  sum  is  16. 

CASE    IV. 

344.  We  will  now  apply  the  principles  exhibited  in  this 
discussion  to  a  few  problems,  which  will  give  rise  to  nearly 
all  the  circumstances  that  usually  occur  in  equations  of  the 
second  degree. 

First  Problem. 

Find  upon  a  line  which  joins  two  luminous  bodies,  A  and 
Bjthe  point  where  these  bodies  shine  with  equal  intensity. 

J^ote. — The  solution  of  this  problem  depends  upon  the 
following  principle  in  physics,  viz. :  The  intensity  of  light 
from  the  same  luminous  body  will  be,  at  different  distances^  in 
the  inverse  ratio  of  the  squares  of  the  distances. 


228  ELEMENTS    OP    ALGEBRA.  [sECT.  VII. 

This  being"  premised,  in  the  indefinite  line  (1,  2)  let  A  and 
B  represent  the  respective  position  of  the  two  lights,  and 
C  the  point  required. 

C  A  C  B  C 

1 1 1 1 1 1 2 

Let   (z=:A  B,  the  distance  between  the  two  lights, 
And  b^=z  the  intensity  of  the  light  A  at  the  unity  distance, 
And  c=  the  intensity  of  the  light  B  at  the  unity  distance. 
Let  a?=r:A  C,  the  distance  from  A  to  the  point  of  equal 

intensity, 
Then  a — a;=B  C,  the  distance  from  B  to  the  point  of  equal 

intensity. 
Then,  by  the  above  principle  in  physics,  the  intensity  of 
A  at  the  distance   1  being  Z?,  its  intensity  at  the  different 

distances  2,  3,  4  ....  a?,  will  be  -,  -,  —  ...._,  which  last 

4   9'  16  s^' 

term  represents  its  intensity  at  C.     In  the  same  manner,  it 

may  be  shown  that  the  intensity  of  B  at  the  distance  a — a?, 

or  at  C,  is  equal  to -.     But  the  conditions  of  the  ques- 

{a — xy 

tion  require  that  their  intensities  be  equal  at  C  ;  hence  we 

have  the  equation 

b  _      0 

x^     {a — xj- 


Reducing         -         x=: ^\/- 


a^h 


Or,  simplifying,       X—  ^        - — I ; 
0 — c 

B  ut  -        l±  's/U= Vb(V'b  ±  v/c). 

And        -        -   b—c=(V'bf—{V'^f={Vb-{-V~c).(V'b—y/~cy, 

'whence.         -         .^-^^M^^±^, 
(Vb-j-VcXVb-Vc) 

Taking   \/c  in  the  numerator,  minus,  and  dividing  the 

equation  by  Vb — \/c,  we  have 

a\/b  * 


(!)• 


V4+Vc' 


SECT.  VIl]  AFFECTED    EQUATIONS.  289 

But,  taking  y/c  in  the  numerator,  plus,  and  dividing  the 
equation  by  v/5-fVc,  we  have 

Hence  we  also  obtain 

a\/c 

(2).  -"^'^ 


Discussion, 
I.    Let  i>c. 

343. — 1.  The  first  value  of  x,  — :r- =,  is  positive,  and  less 

_  Vb-{-Vc  _ 

Vb  ,  ,  s/b 

than  a  ;  for,  — = ;=  being  a  proper  fraction,  a  .  — ^= =<a. 

Vb-{-vc  Vb-^Vc 

This  value  of  a:,  therefore,  gives  the  point  C  between  A  and 

B.     It  is,  moreover,  nearer  B  than  A  ;  for,  in  consequence 

_____  v^6 

of  Z>>c,  we  have  y/b-\-\/b=i2>/b^y/b-{-^c^  whence  —= -=- 

Vb-\-\/c 

1  "^b  ay/b        a      _, 

>i,  and,  consequently,  a . -^-^=^^^>-.    Th.s,  .n- 

deed,  should  be  the  case,  since  we  have  supposed  the  inten- 
sity of  A  greater  than  that  of  B,  or  J>c. 

2.  The  corresponding  value  of  a — a?,  — = =,  is  also  pos- 

Vb-\-y/C 

itive,  and  less  than  ^;  for,  since  j:>-, 

aVb        a 


Vb-^s^c     2 

ay/b 
3.  The  second  value  of  «,  —^- :r,  is  also  positive,  but 

>/b 

greater  than  a  ;  for,  — = being  an  improper  fraction,  a . 

vb — v'c 

20 


230  ELEMENTS    OF  ALGEBRA.  [sECT.  VII. 

-^= --^=— = =:>a.     This  value  of  a?,  therefore,  gives  the 

\/b — -v/c     y/b — \/c 

point  of  equal  intensity  to  the  right  of  B,  at  C    This  should 

evidently  be  the  case,  since  the  light  from  A  and  B  radiates 

in  all  directions.     This  point  will,  moreover,  be  nearer  the 

body,  the  light  of  which  is  least  intense. 


4.  The  second  value  of  a — a?,  — = -^  is  negative,  as  it 

-vb — \/c 

should  be,  since  x^a  ;  and  the  point  C  is  in  the  direction 

opposite  from  A. 

II.  Let  &<c. 

344. — 1.  The  first  value  of  x  will  be  positive,  but  less  than 

-J  for,  v/6  +  V'"^=:2v/^<n/^  +  v/c. 

2.  The  corresponding  value  of  a — x  will  be  positive,  but 
greater  than  -  ;  fox,  since  a?<- 

a\/c        a 
'^~^'^Vb+7c^^' 
Thus,  on  the  present  hypothesis,  the  point  C  will  be  situ- 
ated between  the  two  lights,  but  nearer  A  than  B,  which 
should  evidently  be  the  case. 

3.  The  second  value  of  x.  — = =,  is  essentially  negative, 

and  indicates  that  the  point  of  equal  intensity  is  situated  at 
C",  in  a  direction  from  A  opposite  to  B. 

4.  The  corresponding  value  of  a — x  (which,  since  x  is  es- 

— fl\/c     . 
sentially  negative,  becomes  a+a?),  — = ^,  is  positive  ;  for, 

since  \/c>\/Z>,  the  numerator  and  denominator  will  be  af- 
fected with  like  signs,  and,  consequently,  the  value  of  the 
fraction  will  be  plus.  This  result  also  indicates  that  C" 
should  be  to  the  left  of  A. 

III.  Let  b—c. 
aVb  aVb        aVb     a 


345.— L  The  firt  value  of  x, 


^/^»+Vc     Vb+Vb     2Vb     2' 


8BCT.  VII.]  AFFECTED    EQUATIONS.  231 

ay/c 

2.  The  corresponding  value  of  a — x,  — = =^,  also  equals 

-.     These  two  results  give  the  middle  point  A  B  for  the  first 

required  point,  and  this  result  conforms  to  the  hypothesis. 

as/b 

3.  The  second  value  of  x,  — = =,  since  v/6=\/c,  willbe 

y/h \/c 

reduced  to ,  which  indicates  that  no  finite  value  can  be 

0 

assigned  to  x, 

— as/c 

4.  The  corresponding  value  of  a — a:,  — =; -^  will  also  be- 


come 


y/b—s/C 

— ay/c 


0 

These  results  also  agree  with  the  hypothesis ;  for,  as  the 
diflference  of  their  intensity  decreases,  the  second  values  of 
X  and  a~x  increase,  and,  when  that  difference  becomes  in- 
finitely small,  these  values  must  become  infinitely  large. 
IV.    Let  i=c,  and  a=0. 

346.  The  first  values  of  x  and  a — x  become  0,  and  the 

second  —     This  last  character  is  the  symbol  of  indetermi- 

nation  ;  for,  on  returning  to  the  equation  of  the  problem, 

Q^c)x^—'labx=—a\ 
this  equation  becomes 

O.a:^— O.x==0, 
an  equation  which  may  be  satisfied  by  any  number  whatever 
taken  for  x.  And  this  agrees  with  the  hypothesis ;  for,  if 
the  bodies  have  the  same  intensity,  and  are  placed  at  the 
same  point,  they  will  shine  with  equal  light  upon  any  point 
whatever  in  the  line  1 — 2. 

V.    Let  a=0,  and  b  and  c  unequal. 

347.  Each  of  the  two  values  becomes  0  in  this  case,  which 
proves  that  on  this  hypothesis  there  can  be  but  one  point 
equally  illuminated,  and  that  is  the  point  in  which  the  two 
lights  are  placed. , 


232  ELEMENTS    OF   ALGEBRA.  [SECT.  VII. 

Second  Problem. 
348.  Find  two  such  numbers  that  the  difTerence  of  their 
products  by  the  numbers  a  and  h  respectively  may  be  equal 
to  a  given  number  \y,  and  the  difference  of  their  squares 
equal  to  another  given  number  q. 

Let  X  and  y  represent  the  numbers  sought  ; 
Then        -        -        -        ax — h/=Sj  , 

And  -         -         -         a^ — y^  —q. 

Reducing  these  two  equations,  we  have  for  the  two  values 
of  a?, 


w- 

of 

(2).        - 
The  two  values 

a'-b' 
y  are, 

(1).    . 

bs+aVs'-q{a'-b') 

(2).        - 

bs-aVs'-q(a'-b') 

Discussion. 
I.    Let  a>5. 
349.  In  this  case  a^ — b^  will  be  positive  ;  therefore,  in  or- 
der that  the  values  of  x  and  y  may  be  real,  we  must  have 

q{a'-b')<s\  or  q<. 


a'-b'' 

\.  This  condition  being  fulfilled,  the  first  values  of  x  and 
y  will  necessarily  be  positive,  and,  consequently,  will  form  a 
direct  solution  of  the  problem  in  the  sense  in  which  it  is 
enunciated. 

2.  The  second  value  of  x  will  be  essentially  positive  j  for, 
ayb  gives  asybVs^—q{a'—by  ■"  '^'^'^ 

The  second  value  of  y  may  be  either  positive  or  negative 

In  order  that  it  may  be  positive,  we  must  have 
bs  yas/s'—q{d'—}j')i 

Or,  squaring         -         5V>aV— a'9(a'-Z»') ; 

Or,  transposing     -         bh^-\-a\{a^—b^)>a^s^  j, 


SECT.  VII.]  AFFECTED   EQUATIONS.  233 

Or,  subtracting  W  from  both  sides  of  the  inequation, 

Or,  dividing  -    q>-. 

Thus,  if  a>-,  and  g< — — ,  the  question  is  susceptible 
a*  a* — b^ 

of  a  real  and  direct  solution,  and  will  give  positive  values  of  y. 
But,  if  ?<-j,  and  y<  ^  ,  ^,  the  value  of  y  will  be  nega- 
tive ;  and  we  shall  not  obtain  a  solution  of  the  problem  in 
the  sense  in  which  it  was  enunciated,  but  of  an  analogous 
problem,  the  equations  of  which  are 
ax-\-byz=:s, 

and  which  differ  from  the  proposed  equations  in  this  respect 
only,  that  s  will  express  the  arithmetical  sum  instead  of  the 
difference  of  their  products. 

II.    Let  a<6. 
350.  In  this  case  a^ — b^  will  be  negative,  and  the  values  of 
X  and  y  may  be  put  under  the  form. 


(!)• 


■bVs'+q(b'—a') . 

To  5  ~~  ' 


m                   ^_^as-\-bVs'+q{b^^a^) 
(2).         .         X ^^-^ 

bs-a^^+q{b^-^) . 

(i;.  y jr^^s y 


(2).         .        y= ^     ^, 


_—hs  +  ax/s^-\-q{b^—a!') 

b'—a' 

The  values  of  x  and  y,  it  is  evident,  will  be  real,  since  the 

quantity  placed  under  the  radical  is  essentially  positive. 

Their  first  values  will  be  negative. 

The  second  value  of  y  may  be  either  positive  or  negative. 

s^ 
In  order  that  it  may  be  positive,  we  must  have  q^-^* 

III.   Let  a=b, 
3.t1.  In  this  case  a' — 6*=0,  and  the  first  values  of  x  and  y 
will  be 

Go 


234  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII, 

x=^,  and  y=l^-^. 

The  second  values  of  a?  and  y  will  be 

X—-.  and  y=-. 

But,  if  we  solve  the  given  equations  on  the  hypothesis 
that  a=^,  we  shall  have 

a:=— iJ — ,  and  v=:— ^— — 

The  preceding  discussions  show  the  precision  with  which 
the  algebraic  results  correspond  to  all  the  circumstances  of 
the  enunciation  of  a  problem. 


SECTION  VIIL 

Ratioy  PropQrtion,  and  Progression 

RATIO. 

352.  By  Ratio  is  meant  the  relation  which  one  quantity- 
bears  to  another  with  respect  to  magnitude.  The  quantities 
compared  must  be  of  the  same  nature,  so  as  to  admit  of  a 
common  measuring  unit.  Thus,  we  compare  dollars  with 
dollars,  length  with  length,  weight  with  weight,  time  with 
time,  &c. 

353.  The  magnitudes  of  quantities  may  be  compared  in 
two  ways. 

1.  With  regard  to  their  difference.  This  is  called  jUrith- 
metical  ratio,  or  ratio  by  difference. 

2.  With  regard  to  the  number  of  times  one  quantity  is 
contained  in  the  other.  This  is  called  Geometrical  ratioj  or 
ratio  by  quotient. 

The  Arithmetical  ratio  of  two  numbers,  as  a  and  bj  is  ex- 
pressed, a — hy  or  a  .  .b. 


SECT.  VIII.]  RATIO.  235 

The  Geometrical  ratio  of  two  numbers,  us  a  and  i,  is  ex- 
pressed, aiby  or  -. 

0 

When  the  ratio  is  thus  expressed,  the  first  term  is  called 
the  antecedent^  the  last  term  the  consequent,  and  the  two  terms, 
takin  together,  are  called  a  couplet, 

354-.  The  term  arithmetical  ratio  is  only  a  substitute  for 
the  word  difference,  and  involves  no  principle  that  is  not  es- 
sentially involved  in  algebraic  subtraction. 

355.  In  a  geometrical  ratio  three  things  are  involved,  viz., 
the  antecedent,  the  consequent,  and  the  ratio ;  and  any  two  of 
these  being  given,  the  other  may  be  found. 

Let  a=  antecedent,  c=  consequent,  and  r=  ratio  : 
Then,  from  the  geometrical  ratio  a:  c=r,  we  have 

r=-,  i,  e.,  ratio  =  antecedent  -r-  by  consequent ; 

And,  a=c .  r,  i.  e.,  antecedent  =  consequent  x  by  ratio  j 

And,  c=-,  i.  c,  consequent  =  antecedent  -i-  by  ratio. 

T 

356.  When  the  antecedent  is  equal  to  the  consequent,  the 
ratio  is  a  unit,  or  a  ratio  of  equality.  When  the  antecedent 
is  greater  than  the  consequent,  the  ratio  is  greater  than  a 
unit,  or  a  ratio  of  greater  inequality.  When  the  antecedent 
is  less  than  the  consequent,  the  ratio  is  less  than  a  unit,  or 
a  ratio  of  less  inequality. 

A  compound  ratio  is  the  ratio  formed  by  multiplying  the 
corresponding  terms  of  two  or  more  ratios. 

A  duplicate  ratio  is  the  ratio  of  the  squares  of  the  corre- 
sponding terms  of  a  ratio  ;  the  triplicate  ratio,  of  the  cubes 
of  the  corresponding  terms  j  the  sub-duplicate,  of  the  square 
roots  of  the  corresponding  terms,  &c. 

357.  The  ratio,  it  will  be  observed,  is  expressed  by  a  frac- 
tion, the  antecedent  becoming  the  numerator  and  the  conse- 
quent the  denominator.  Now  it  has  been  demonstrated  that, 
if  both  terms  of  the  fraction  be  multiplied  or  divided  by  the 


236  ELEMENTS    OF   ALGEBRA.  [SECT.  VIII. 

same  number,  the  value  of  the  fraction  will  not  be  affected. 
Hence  we  infer, 

1.  If  the  terms  of  a  ratio  he  multiplied  or  divided  hy  the 
same  number,  it  does  not  alter  the  value  of  the  ratio. 

2.  j1  ratio  may  be  reduced  to  its  lowest  terms  by  dividing  its 
antecedent  and  consequent  by  their  greatest  common  measure. 

3.  Ratios  may  be  compared  with  each  other  by  reducing  the 
fractions  which  represent  their  values  to  equivalent  fractions 
having  a  common  denominator. 

358.  The  following  are  some  of  the  more  important  theo- 
rems relating  to  ratios  : 

1.  A  ratio  o{  greater  inequality  is  diminished,  and  a  ratio  of 
lesser  inequality  is  increased,  by  adding  the  same  quantity  to 
both  members. 

First.  Let  a+bia,  or  ^^^—-,  represent  a  ratio  of  greater 
a 

inequality : 

Adding  x  to  both  terms     -    a-\-b-i-x:a-^x,  or —^  j 

a-\-x 

Then         -         -         -         -     a-\-b:aya^b-\-x:  a-{-Xy 
Or     -         -         -         -         -         a+b     a-\-b-{-x  ^ 

a  a-^x 

For,  reducing  to  a  ?  a'^-\-ab-\-ax-^bx^a^-{-ab-{-ax 
common  denom.  >         a{a-\-x)  a(a-\-x) 

Second.  Let  a — b :  a,  or  ,  represent  a  ratio  of  lesser 

a 

inequality : 

fi h    I    /P 

Adding  x  to  bath  terms    -    a — b-\-x :  a-\-x,  or J— ; 

a-\-x 

Then         .        -         -         >    a — b:a<Ca — b-\-x:a-\-x, 

.^  a — b  ^a — b-\-x 

^^ < r J 

a  a-{-x 

For,  reducing  to  a  j>  a^-j-ax — ab — bx^a^ — ah-\-ax 


\        a( 


common  denom.  )         a(a-f  ^)  a{a-\-x^ 

2.  A  ratio  of  greater  inequality  is  increased:;  and  a  ratio  of 
lesser  inequality  is  diminished,  by  subtracting  the  same  quan- 
tity from  both  terms. 


SECT.  VIII.]  RATIO.  237 

First.  Let  a+h:ay  or  y^,  represent  a  ratio  of  greater 

a 

inequality  : 
Subtracting  X  from  both  terms,  a+h—X'.a-^x^ox—l- ; 


Then         -        -        -        -      a-\-hia<ia-{-h^x'.a—x^ 

Or !Lh^<^±^; 

a  a—x 

For,  reducmg    -  — ^— < — —  . 

I  a{a — x)  a[a—x) 

Second.  Let  a+Jio,  or  ^^^,  represent  a  ratio  of  lesser 
a 

inequality  : 

Subtracting  x  from  both  terms,  a— J — x :  a — x,  or H- ; 

a — X 

Then         -         -        -        -      a — bia^a — b — x:a — x, 

Or g-j^g-i-x. 

a  a — X 

«  J     •  a* — ab — ax+bx^a^ — ab — ax 

For,  reducmg    -  1 — > 

a{a — x)  a{a — x) 

3.  A  ratio  of  greater  inequality  compounded  with  another 
ratio  increases  it  ;  but  a  ratio  of  lesser  inequality  compound- 
ed with  another  ratio  diminishes  it. 

First.  Let  a+bia^  or  ^— -,  represent  a  ratio  of  greater 

inequality, 

And  -         -         -         -      TO  : »,  or  -,  be  any  other  ratio  : 

n 


Compounding    -         -      am-{-bm  :  an^  or  ^    "^ _  j 

an 

Then  ...         m:n<iam-{-bm:anf 

^  m^a7n-\-bm  ^ 

n         an 


For,  reducing    .        -  <m»    amn+*m» 

an*  an^ 

Second.  Let  a—b:a,  or  — H-,  represent  a  ratio  of  lesser 


an'  an" 

a-l 
a 
inequality, 


238  ELEMENTS    OF    ALGEBRA.  [SECT.  VIII. 

And  -        -        '•        -      m:n,  or  —,  another  ratio  : 

n 


ri                J-                                     I                    «^ — ^wi 
Compounding    -         -      am — om  :  an^  or j 

Then  -         -         -        min^am — hmian, 

n                                                 ^^  ctm — hm 
Or -> j 

n  an 

■ry  J     •  amn^  amn — hmn 

For,  reducing    -        -  — -  > 

avr  ainr 

4.  If  to  the  terms  of  any  couplet  there  he  added  two  other  terms 
having  the  same  ratio,  the  sums  will  have  the  same  ratio. 

Let  the  ratio  a :  b  equal  the  ratio  c:d: 

Then         -    W'        •        '        r-^,=T=-,  5 
r  b+d     b     d 

For,  since  -        -         -        -        a  :  6=c  :  c?. 

We  have  -----     -=-. ; 

b     d 

Clearing  of  fractions  -         -         -   adz=zbc  ; 

Adding  cd  to  both  members,  ad-\-cd—hc-\-cd; 

Resolving  into  factors        -    d{a-\-c)=:c{b-{-d)y 

Dividing   -         -         -         -        -^_=    =   . 

b+d     d     b 

5.  If  from  the  terms  of  any  couplet  two  other  quantities  hav-^ 
ing  the  same  ratio  be  subtracted,  the  remainders  will  have  the 
same  ratio. 

Let  the  ratio  a :  b=  the  ratio  c:  d: 

Then  .  -  -  -  - 
For,  since  -  -  -  -  - 
"We  have 


a- 
a- 

~c 
-d" 

a     c 
'-b—d' 

a 

:  5= 

-.c:d, 

a_ 

b~ 

c 

ad. 

.Ic; 

Clearing  of  fractions 
Subtracting  cd  from  both  members,  ad — cd=bc — cd; 
Resolving  into  factors  -  -  c?(a — c)=c(b — d); 
TV'     !•  a — c     c     a 


SECT.  VIII.]  PROPORTION.  ,  239 

EXAMPLES. 

1.  Reduce  the  ratios  360 :  315,  and  1595 :  667,  to  their  low- 
est terms. 

2.  Which  is  the  greater  of  the  two  ratios,  11:9,  and 
44:35] 

3.  Which  is  the  least  of  the  three  ratios,  20 :  17,  22 :  18, 
and  25 :  23  1 

4.  If  the  consequent  be  35,  and  the  antecedent  985,  what 
is  the  ratio  1 

5.  If  the  antecedent  be  1512,  and  the  ratio  12,  what  is  the 
consequent  1 

6.  If  the  consequent  be  320,  and  the  ratio  i,  what  is  the 
antecedent  1 

7.  What  is  the  compound  ratio  of  12 :  21, 18 :  6,  and  24 : 5 1 

8.  What  kind  of  a  ratio  will  be  produced  by  compounding 
5x4-7  :  2«— 3,  and  x-\-2  :  ix-{-3  1 

9.  What  kind  of  a  ratio  will  be  produced  by  compounding 
c^—3^ :  a',  a-\-x  :  h,  and  b  :  a—x  ? 

10.  What  kind  of  a  ratio  will  be  produced  by  compound- 
ing x-\-y :  a,  x—y  :  ft,  and  b :  — H^  % 

11.  What  is  the  ratio  produced  by  compounding  3 :  7,  the 
duplicate  ratio  of  3 : 5,  and  the  triplicate  ratio  of  4  :  3  1 

12.  What  is  the  ratio  produced  by  compounding  the  sub- 
duplicate  ratio  of  49 :  4,  and  the  sub-triplicate  ratio  of  64 : 
1251 

PROPORTION. 

359.  Ratio  is  a  comparison  of  two  quantities  to  ascertain 
their  difference,  or  how  often  one  is  contained  in  the  other. 

Proportion  is  a  comparison  of  two  equal  ratios. 

If  the  ratios  are  arithmetical,  the  proportion  is  called 
arithmetical  proportion^  or  proportion  by  difference. 

If  the  ratios  are  geometrical,  the  proportion  is  called 
geometrical  proportion^  or  proportion  by  quotient. 

360.  There  are  always  two  couplets,  or  four  terms,  in  a 
proportion.     The  first  and  fourth  terms  are  called  extremes  ; 


240  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII. 

the  second  and  third,  means.  The  two  antecedents,  or  the 
two  consequents  taken  together,  are  called  homologous  terms. 
The  terms  of  the  same  couplet  are  called,  with  reference  to 
the  proportion,  analogous  terms.  Three  terms  are  said  to  he 
proportional  when  the  ratio  formed  by  the  first  and  second 
is  equal  to  the  ratio  formed  by  the  second  and  third. 

361.  As  an  arithmetical  proportion,  or  a  proportion  by  dif- 
ference^ is  nothing  more  than  a  simple  form  of  equation,  it  is 
unnecessary  to  give  the  subject  a  separate  consideration. 
It  is  expressed  a— 6— c— c/,  or  a . .  &=c  . .  d. 

362.  Geometrical  proportion  is  expressed  by 
*  a:b=zc:d, 

Or         -         a:b::c:dj 
which  expressions  are  read,  "  a  to  b  equals  c  to  c?,"  or,  "  a 
is  to  &  as  c  to  d" 

THEOREMS  RELATING  TO  PROPORTION. 
363. — (1.)  If  four  numbers  be  proportional,  the  product  of 
the  extremes  will  be  equal  to  the  product  of  the  means. 
Let         -         -         -         -         a:b'.',c:d;f 

Then,  by  equality  of  ratios,        _=-  j 

0  (i 

•Clearing  of  fractions      -         •  ad=:zbc. 

CoR.  1.  Any  factor  may  be  transferred  from  one  mean  or 
extreme  to  the  other  without  destroying  the  proportion. 
Thus,  \{  a:b::  cmidn,  then  an:bm::c:d. 

CoR.  2.  If  any  three  terms  of  a  proportion  be  given,  the 
fourth  can  always  be  ascertained  ;  for  if  a :  & : ;  c :  cZ, 

Then      -         -   ad=cb ; 

cb 
"    Dividing  by  d,     a=— ,  i.  c,  the  1st  termr=2dx3d-^4th; 
d 

Dividing  by  c,     b=—,  i.  e.,  the  2d  term  =  1st  x  4th -^  3d  ; 
c 

Dividing  by  b,     c=—,  i.  e.,  the  3d  term=:lstx4th-f-2d ; 
b 

Dividing  by  a,     d=z^—,  i.  e.,  the  4th  term=2dx  3d-T-lst. 
a 


8«CT.  VIII.]  PROPORTION.  241 

364. — (2.)  If  tkt  product  of  any  two  numbers  he  equal  to  the 
product  of  two  others^  these  four  numbers  will  constitute  a  pro- 
portion  when  so  arranged  that  th^f actors  of  one  product  be  made 
the  meanSy  and  the  factors  of  the  other  product  the  extremes. 

Let ad=bc: 

Dividing  by  db,  and  reducing,      -=-  j 

b     d 

Hence    -         -        -        -         a:b::c:d, 

365. — (3.)  If  three  numbers  be  propoitional,  the  product  of 

the  two  extremes  is  equal  to  the  square  of  the  mean. 

Let         -         -         -         -         a:b::b:c : 

Then ?=*, 

b     c 

Or acz^lr". 

Cor.  The  mean  proportional,  or  geometrical  mean,  be- 
tween two  numbers  is  equal  to  the  square  root  of  their 
product.     Thus,  if  ac^b"^,  then  bz=y/ac. 

366. — (4.)  If  four  numbers  be  proportional^  1.  the  order  of 
the  extremes^  2.  of  the  means,  3.  of  the  terms  of  each  couplet,  4. 
of  the  couplets,  5.  of  all  the  terms,  may  be  inverted  withoitt  cfe- 
stroying  the  proportion. 

Let  .      -        -         -  a'.b'.'.c'.d: 

Then     ....       t='-. 

b     a 

Dividing  by  a,  and  multiply  )  d^c^  ,-.  d:b::c:a  ;     (1.) 

ing  hy  d         -         -         -     S  b     a 
Dividing  by  c,  and  multiply-  )  a_b^  .-.aiciibid;      (2.) 

ing  by  6  ---)crf* 

Inverting  the  fractions       -       -=-»  •'•  h:a::d:c ;      (3.) 

a     c 

Inverting   the   order  of  the  )  c_a    .   -  .  ^ . .     .  i  .      /a\ 

members        •         -         -     }d     V 
Inverting  the   fractions  and  j  ,     , 

changing  the  order  of  the  > ~~~,  .'•  d:c::b:a,       (5.) 

terms     -        -         -         -     ) 

367. — (5.)  If  four  numbers  be  proportional,  the  ^dnalogous 
21  Hh 


242  ELEMENTS    OP    ALGEBRA.  [sECT.  VIII. 

or  the  Homologous  terms  may  be  multiplied  or  divided  hy  the 
same  number  without  destroying  the  proportion. 
Let        -        -        -        -       a:b::c',d: 

Then "L=t 

b     d 

Multiplying  the  terms  of  the  }  am     c  ,  , 

n       r       •      X  {  -,—  =  -,»  '''  am  :  bm  :  :  c  :  d  : 

nrst  traction  by  m  -         -        S  bm     d 

Multiplying  the  terms  of  the  }     a     cm  ,  , 

^•^     °  [    -=—,.'.a:b::cm:dm; 

second  traction  oy  m      -        )     b     dm 


am     cm 


Multiplying  the  equation  hy  wz,    ==: , ,-,  am  :  b  :  :  cm:  d  ; 

b       d 

Dividing  the  equation  by  m,         J^=_^,  .'.a  :  bm  :  :  c  :dm ; 

bm     dm 


Dividing  the  terms  of  the  first  1  -^      c       a     b 
fraction  hy  m  -         -         -        \   ^~d^ '  '  m'  m 


c  :  d ; 


Dividing  the  terms  of  the  sec-  )    ^     ^  c      d 

i     6     1'    •  mm 

J  m 


end  fraction  by  m 


a     ,        c 


Dividing  the  numerators  hy  m  -  21= ul,  »'*  —  i  b  :  :  —  :  dj 

b     d       m       '     m 

Dividing  the  denominators  by  w,  x^T")  •*.«:—  :  :  c  :  _. 

mm  ^  m 

368. — (6.)  If  there  be  two  sets  of  proportions  having  an  ante- 
cedent and  consequent  in  the  one  equal  to  an  antecedent  and  con- 
sequent of  the  other,  the  remaining  terms  will  be  proportional. 


Let    -        -        -        - 

a  : 

:b::c:d, 

And   - 

a  : 

b :  :  m: n: 

By  the  first  proportion  - 

- 

a     c 

■b-r 

By  the  second 

- 

a     m 

Therefore       - 

- 

f =^,  and  c  :  rf 
d     n 

:  n. 


369.— (7.)  If  two  homologous  or  two  analogous  terms  be  added 
to  or  subtracted  from  the  two  others,  the  proportion  will  be  pre- 
served. 


SECT.  VIII.]  PROPORTION.  243 

FirBt,  Let      -        -        -        aib  i\  ci  d^ 
r\  O      C 

^'   •    •    ■    -  i=r 

Then  (by  Art.  358,  th.  5),  i±^=?=^  j 

bztd     b     d 

Hence   -        -        -    a-\-c  :  b-\-d  ::  a  :  b^  or  as  c  :  d, 

And        -         -         -     a — c  :  b — d  ::  a  :  b^  or  as  c  :  d. 


Second,  Inverting 

the  order  of  the  means,  a  :  c  :  :  b  :  d^ 

Then     - 

a±b    a    b  . 
c±d    c     d* 

Hence    - 

a-\-b  :  c-^d  ::  a  :  c,  or  as  b  :  d^ 

And       - 

a—b  :  c—d  :  :  a  :  c^  or  as  b  :  d. 

Cor.  1.  Since 

a±i:_a  ,   a-f  c     a — c  . 
b±d     b      b-\-d     b—d' 

Hence  we  have 

a  +  c  :  a—c  :  :  b-\-d  :  b — d. 

Cor.  2.  Since 

a±b_a  ,   a-\-b_a — A. 
c±d     b      c-\-d     c—d' 

Hence  we  have 

a-{-b  :  a — b  :  :  c-\-d  :  c — d. 

370. — (8.)  If  two  sets  of  proportional  numbers  be  multiplied, 
the  products  of  the  corresponding  terms  will  be  proportional. 

Let     -         -         -     a  :  b  :  :  c  :  dy  or  -=-=. 

b     d 

And    -         -         -     m  :  n  :  :  p  lOyOT  —  =?■ : 

n     q 

Multiplying  the  two  equations     -     ^=^  ; 

bn     dq 

Hence    -         -         -         -     am  :  bn  :  :  cp  :  dq. 

37  L — (9.)  If  one  set  of  proportional  numbers  be  divided  by 
the  corresponding  terms  of  another  set,  the  quotients  will  be  pro- 
portional. 

Let     -         -         a  :  ^  :  :  c  :  (/,  or  ^=-p 

b     d 

And    -  mm:  :  p  :  q,  or  -  =?■ : 

n     q 

Dividing  the  first  equation  by  )  a  .  m     c^p 

the  second  -        -        -         }  b     n     d     q* 

T     ^'      m    n      p    q 


244  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII. 

372. — (10.)  If  four  numlexs  he  proportional^  like  powers  or 
like  roots  of  them  will  be  proportional. 
Let     -         -         a  :  h  :  :  c  :  d: 


Then 


a_c  , 
b~d' 


Involving       -         -     —  =-t  »  •*•  a""  :  Jf"  :  :  c^  :  d""  / 

^  (1       v/c  —  —  

Evolving         -         -  -==-=,  .-.  ^a:  :yb::  V'c:  ^d. 
Vb     \/a 

PROBLEMS  TO  BE  SOLVED  BY  PROPORTION. 

1.  There  are  two  numbers  whose  difference  is  to  the  l«ss 
as  100  is  to  the  greater,  and  the  same  difference  is  to  the 
greater  as  4  to  the  less.-    What  are  the  two  numbers  1 

Let  x=  the  greater,  and  y=  the  less : 
Then  -  -  -  -  -  x — y  :  y 
'  And  .  -  -  -  .  cc — y  :  x 
Multiplying  the  proportions,  (x — yY  :  xy 
Dividing  consequents  -  (x — yY  :  1 
Evolving  ...  -  X — y  :  1 
Converting  into  an  equation  -  x — y=20  ; 
Whence  -         -         -         -  x—^0-\-y; 

Substituting  this  value  of  x  )  ^O+y-y  :y::  100  :  20-f  y, 

in  the  first  proportion     ) 
Or           -         -         -         -        20  :  y  :  :  100  :  20  +  y; 
Dividing  antecedents      -          1  :  y  :  :       5  :  20+y  ; 
Convertiag  into  an  equation  -        5y=20+2/  ; 
Whence          .         -         -         -  y=5, 

And a?=20  +  5  =  25. 

2.  The  product  of  two  numbers  is  15,  and  the  sum  of  their 
squares  is  to  the  difference  of  their  squares  as  17  to  8.  What 
are  the  numbers  1 

Let  x=  the  greater,  and  y=  the  less : 


100 

X, 

4 

y; 

400 

xy; 

400 

1; 

20 

ij 

Then 

. 

xy—16, 

And 

. 

x^-\-f  :  x'-y'  : 

r  17:  8; 

Adding 

y  and  subtracting  - 

2x'  :  2f  : 

:  25  :  9  ;. 

4 

SECT.  VI [1.1 


PROPORTION. 


246 


Dividing  first  couplet  by  2 
Evolving        -        -         - 
Whence  -         -         . 

Reducing       -        -        - 


-  x'  :  y«  :  :  25  :  9  ; 

-  X  : y  :  :    5:3; 
ar=5y  ; 

a:=5,  and  y=3. 


3.  What  two  numbers  are  those  whose  product  is  320, 
and  the  difference  of  their  cubes  is  to  the  cube  of  their  dif- 
ference as  61  to  11 

Let  xz=i  the  greater,  and  y=  the  less : 

Then a?y=z32a, 

And  -----  x^—y^  :  (x—yY  : 
Expanding  2d  term,  a?*— y*  :  x^—3j^y-\-3xy^—y^  : 
Subtracting  consequents  ^02 

from  antecedents         -  )        ^ 
Dividing  first  ratio  by  x— y     - 
Dividing  antecedents  by  3 
Substituting  value  of  Jry  - 
Dividing  antecedents  by  20    - 
Evolving         -         -         -         - 
Converting  into  an  equation   - 
Whence         -         -         -         - 

4.  It  is  required  to  prove  that  a  :  x  :  :  V2a — y  :  v/y^  on 
the  supposition  that  (a-fx)*  :  (a—xf  :  :  x-f-y  :  x— y. 

Expanding  first 
and  2d  terms, 
•Adding  and  subtracting,  2o'4-2x* :  4ax  : 

Dividing  terms       -         -  a^-\-  x^ :  2ax  : 

Transferring  the  factor  x,        o'-f-  x* :  2a    : 
Inverting  means     -         -  a*-f-  x* :  x*     ; 


-3xy»  :  (x-yf  : 

3xy  :  (x-yf  : 

xy  :  (x—yY  : 

320 :  (x-yY  : 

16  :  {x-yf  : 

4  :  X — y  : 

X— y=:4 ; 

-  x=20,  and  y 


:61. 

■•  J 

:  61  • 

^  5 

:60 

■'■  5 

:60 

■^  > 

:  20 

1  y 

:20 

1; 

:    1 

1 5 

:     1  : 

*  J 

16. 


''  i  a2+2ax+x« :  a*— 2ax-|-x* :  :  x+y  :  x— y; 

2x :  2y  ; 


a':x» 


a  :  X 


X  :y; 
x^ry; 
2a:y; 
2a— y:y; 


Subtracting  terms  - 
Evolving        -         -         - 
5.  It  is  required  to  prove  that  dx—cy^  on  the  supposition 
that  X  :  y  :  :  a^  :  b\  and  a  :  1 


V2a—y:y/y, 


Vc+x 


^d+y. 
Inverting  the  order  of  the  ratios  >  _,     ,, 

m  the  first  proportion  -    3 

Involving  second  proportion  -       a'  :  d'  :  :  c-\-x  :  d+y ; 
By  equality  of  ratios      -         -       x  :y  :  :  c-\-x  :  d-\-y; 


246  ELEMENTS    OF    ALGEBRA.  [&ECT.  VIII. 

Inverting  means     -*       -         -       x  :  c-\-x  :  :  y  :  (i-\-y ; 
Subtracting  terms  -         -         -       x   :  c         :  :  y  :  d  ^ 
Converting  into  an  equation  -  dx=:cy. 

6.  There  are  two  numbers  whose  product  is  24,  and  the 
difference  of  their  cubes  :  the  cube  of  their  difference  as 
1&  :  1.     What  are  the  numbers  1  j^ns.  6  and  4. 

7.  The  sum  of  two  numbers  is  to  their  difference  as  3  :  1, 
and  the  difference  of  their  third  powers  is  56.  What  are 
the  numbers  %  Ans.  4  and  2. 

8.  There  are  two  numbers  whose  product  is  135,  and  the 
difference  of  their  squares  is  to  the  square  of  their  differ- 
ence as  4  to  1.     What  are  the  numbers'?     Ans.  15  and  9. 

9.  There  are  two  numbers  which  are  to  each  other  in  the 
duplicate  ratio  of  4  to  3,  and  24  is  a  mean  proportional  be- 
tween them.     What  are  the  numbers  %      Ans.  32  and  18. 

10.  Tl^ere  are  two  numbers  which  are  to  each  other  as  3 
to  2.  If  6  be  added  to  the  greater  and  subtracted  from  the 
less,  the  sum  will  be  to  the  remainder  as  3  to  1.  What  are 
the  numbers'?  Ans.  24  and  16. 

11.  What  number  is  that  to  which  if  3,  8,  and  17  be  sev- 
erally added,  the  first  sum  will  be  to  the  second  as  the  sec- 
ond to  the  third  1  Ans.  3i. 

12.  The  sum  of  the  third  powers  of  two  numbers  is  to  the 
difference  of  the  third  powers  as  559  to  127,  and  the  square 
of  the  first,  multiplied  by  the  second,  is  equal  to  294.  What 
are  the  numbers  %  Ans.  7  and  6. 

ARITHMETICAL  PROGRESSION. 

373.  A  series  of  numbers  increasing  or  decreasing  by  a 
constant  difference,  is  called  an  arithmetical  progression^  or 
progression  by  difference. 

374.  When  the  numbers  increase  by  a  common  differ- 
ence, they  form  an  ascending  series  /  when  they  decrease,  a 
descending  series. 

Thus,  the  natural  numbers, 

1,  2,  3,  4,  5,  6,  7,  8,  &c.,  ' 
form  an  ascending  series. 


2.  Transposing,  &c.,    a=l — {n — 1)(/,  i 


BECT.  VIII.]  ARITHMETICAL    PROGRESSION.  247 

Inverted,  they  form  a  descending  series  j  as, 
8,  7,  6,  5,  4,  3,  2,  1. 

375.  From  the  definition  of  arithmetical  progression,  it  is 
evident  that  in  an  ascending  series  each  term  is  found  hy 
adding  the  common  difference  to  the  preceding  term. 

Let  a=  first  term,  (f=  common  difference,  and  n=  the 
number  of  terms  : 

Then  the  terms  of  the  series  will  be 
12         3          4          5  n 

c,  a+d,  a4-2J,  a-f  3(f,  a-\-4>d a-\-(n^l)d. 

Hence,  letting  /=  the  last  term,  we  shall  have, 

-  I        I  /       i\ ;  ?  the  formula  for  the 

1.        -        -        -         lz=za-{-(n — l}a,  \ 

^         '     >      last  term. 

the  formula  for  the 

first  term. 

3.  Transposing  and  >    ,_/ — a  >  the  formula  for  the 
dividing  -         -     )         n — l'            )      common  diff. 

.    rr  .        .  / — a  .  -        >  the  formula  for  the 

4.  Transposing,  &c.,    n  =  — r-+l,       J  ,        ^ 

d  3     number  of  terms. 

These  four  formulas  may  be  enunciated  as  follows  ; 

1.  The  last  term  is  equal  to  the  first  term^  plus  the  common 
difference  multiplied  by  the  number  of  terms  less  one. 

2.  The  first  term  is  equal  to  the  last  term,  minus  the  common 
difference  multiplied  by  the  number  of  terms  less  one. 

3.  The  common  difference  is  equal  to  the  difference  between  the 
extremes  divided  by  the  number  of  terms  less  one. 

4.  The  number  of  terms  is  equal  to  the  difference  between  the 
extremes  divided  by  the  common  difference,  the  quotient  increased 
hy  one. 

376.  If  the  series  is  descending,  the  above  formulas  wijl 
evidently  become, 

1.  -         l=a—(n—l)d; 

2.  -        a=t+(n—l)d; 

n — 1 
a 


•248  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII. 

377.  If  the  common  difference  and  first  term  are  equal, 
l=za-{-(n — l)d=a-{-(n — l)a=a-\-an — a=any 

Or,  l=a — (n — l)c?=a — (n — l)a=a — an-\-a  —  2a—an. 

378.  From  the  third  formula,  d=       ^,  we  may  obtain  a 

n — 1 

general  method  for  finding  any  number  of  arithmetical 
means  between  two  given  numbers.  To  do  this,  it  is  only 
necessary  to  obtain,  in  addition  to  the  given  data,  the  com- 
mon difl^erence. 

Let  m~  the  number  of  means.  Then,  since  the  whole 
number  of  terms  consists  of  two  extremes,  plus  the  means,  we 
shall  have  m-\-'2.=n. 

Hence,  substituting  for  n  its  value  in  the  above  formula,, 

1 / — a     / — 'a 

~m^-2—l~m-^l 

PROBLEMS  FOR  SOLUTION. 

1.  The  first  term  of  an  arithmetical  progression  is  50,  and 
the  common  difference  10.     What  is  the  100th  term  1 

Ans,  /=a-h(7i—l)c/3=:50+(100—l).  10=1040. 

2.  The  first  term  of  an  arithmetical  series  is  120,  and 
the  common  difference  2.     What  is  the  325th  term  ] 

Jlns. 

3.  The  first  term  of  an  arithmetical  series  is  2,  the  last 
term  1828,  and  the  number  of  terms  42.  What  is  the  com- 
mon difference  'X  Ans. 

4.  The  last  term  of  an  arithrhetical  series  is  2680,  the 
common  difference  5,  and  the  number  of  terms  30.  What 
is  the  first  term  1  Ans, 

5.  The  first  term  of  an  arithmetical  series  is  8,  the  last 
term  1728,  and  the  common  difference  2.  What  is  the 
number  of  terms  1  '  Ans. 

6.  The  first  term  of  a  decreasing  arithmetical  series  is 
800,  the  number  of  terms  21,  and  the  common  difference  2. 
What  is  the  last  term  %  Ans, 

7.  Find  4  arithmetical  means  between  2  and  52. 

Ans,  ci=10  J  and  the  series,  2,  12,  22,  32,  42,  52. 


SECT.  VIII.]  ARITHMETICAL    PROGRESSION.  349 

8.  The  first  term  of  a  descending  arithmetical  series  is 
480,  the  last  term  12,  and  the  number  of  terms  42.  What 
is  the  common  difference  \  Ans. 

9.  Find  8  arithmetical  means  between  12  and  52920. 

Ans.  d=     ,  and  the  series 

10.  The  first  term  m  a  descending  arithmetical  series  is 
46450,  the  last  term  10,  and  the  common  difference  2.  What 
is  the  number  of  terms  ]  Ans. 

SUM  OP  THE  SERIES. 
379.  The  sum  of  the  series  may  evidently  be  obtained  by 
the  addition  of  all  the  terms,  nor  will  this  sum  be  affected 
if  the  order  of  the  terms  be  inverted.     Thus, 

S=[a  ]-h[a  +  (iJ-h[a+2c/]  +  [a4-3c/]-f  ■  -  -  +[«+ 

^n^4,)d]  +  [a-\-{n^3)d)-\-[a-^{n—2)d']-\-[a-\-(n—l)d]i 

S=[a-f.(n— l)rf]  +  [a-f(»-2)(i]-h[a+(«— 3yj  +  [a+(/i— 

4)(/]-f  .  -  .  4- [a-h3(f]  +  [a-f2(i]  +  [a-}-rf    ]-f  [a  ]. 

Adding  the  two  equations, 

2S=[2a-|-(»— l)(/]-h[2a+(»— l)(i]4-[2a-h(«— iy]  +  [2a 
+  {n—l)d]+  .  .  -  -f  [.2a+(;i-l)c/]  +  [2a+(n— l)c/}+[2 
.        a-^(^ri—l)d]  +  [2a  +  {n—l)d]. 
But,  since  there  are  n  terms,  and  all  the  terms  are  equa>, 

2S=[2a+{n-i)d]n=. 
Hence,  by  performing  the  necessary  reductions, 
1    o_2a4-(» — l)<^w      )  ti^e  formula  for  the  sum  of  the 
2  )      terms. 

2.  a= ,  the  formula  for  the  first  term. 

2/1 

3    d=.^~l ,  the  formula  for  the  common  difference. 


y/{2a—d)^-\-Hds—2a-\-d  ^  the  formula  for  the  num- 


'A 


2d  )      ber  of  terms. 

These  four  formulas  may  be  enunciated  in  tho  form  of 
general  propositions  or  rules. 

PROBLEMS  TO  BE  SOLVED  BY  THE  PRECEDING 
FORMULAS. 
1.  The  first  term  of  an  arithmetical  series  is  5,  the  num. 

Ii 


*^0  ELEMENTS    OF   ALGEBRA.  [sECT.  VIII. 

ber  of  terms  30,  and  the  common  difference  3.    What  is  the 
sum  of  all  the  terms  1  Ans.  1455, 

2.  The  sum  of  the  terms  of  an  arithmetical  series  is  280, 
the  first  term  1,  and  the  number  of  terms  32.  What  is  the 
common  difference  1  Ans.  |. 

3.  The  sum  of  the  terms  of  an  arithmetical  series  is  950, 
the  common  difference  3,  and  the  number  of  terms  25.  What 
is  the  first  term  %  Ans.  2. 

4.  Suppose  100  balls  be  placed  in  a  straight  line,  at  the 
distance  of  a  yard  from  each  other  ;  how  far  must  a  person, 
starting  from  the  box,  travel  to  bring  them  one  by  one  to  a 
box  placed  at  the  distance  of  a  yard  from  the  first  ball  \ 

Ans.  5  miles  and  1300  yards. 

5.  In  gathering  up  a  certain  number  of  balls,  placed  on 
the  ground  in  a  straight  line,  at  the  distance  of  2  yards  from 
each  other,  the  first  being  placed  2  yards  from  the  box  in 
which  they  were  deposited,  a  man,  starting  from  the  box, 
travelled  11  miles  and  840  yards.  How  many  balls  were 
there  1  Arts.  100. 

6.  How  many  strokes  do  the  clocks  of  Venice,  which  go 
on  to  24  o'clock,  strike  in  a  day  \  Ans.  300. 

7.  In  a  descending  arithmetical  series  the  first  term  is 
730,  the  common  difference  2,  and  the  last  term  2.  What 
is  the  number  of  terms  1  Ans.  365. 

8.  A  speculator  bought  47  house  lots  in  a  certain  village, 
giving  $10  for  the  first,  $30  for  the  second,  $50  for  the 
third,  and  so  on.     What  did  he  pay  for  the  whole  47  1 

Ans.  $22,090. 

9.  A  man  bought  a  certain  number  of  acres  of  land,  pay- 
ing for  the  first  $i,  for  the  second  $|,  and  so  on.  When 
he  came  to  settle,  he  had  to  pay  $3775.  How  many  acres 
did  he  purchase,  and  how  much  did  he  give  per  acre  1 

Ans.  150  acres,  at  $25^-  per  acre. 

10.  A  wealthy  gentleman  offered  to  his  daughter,  on  the 
evening  of  her  marriage,  $50,000  as  her  dowry  ;  or  he  would 
give  her  on  that  evening  $1,  on  the  next  $2,  and  so  on  to 


SECT.  VIII.]  GEOMETRICAL    PROGRESSION.  251 

the  end  of  the  year,  365  days,  and  also  the  balance  of  inter- 
est that  might  be  found  in  her  favour  if  she  accepted  the  lat- 
ter offer.  The  lady,  being  unskilled  in  mathematit;s,  chose 
the  first  ofier.     Did  she  gain  or  lose  by  this  choice  1 

Jlns.  She  lost  $16,795. 

GEOMETRICAL  PROGRESSION.  / 

380.  If  a  series  of  numbers  increase  or  decrease  by  the 
continued  multiplication  or  division  by  the  same  number, 
they  are  said  to  be  in  Geometrical  Progression,. 

381.  When  the  numbers  increase  by  a  common  multiplier, 
they  form  an  ascending  geometrical  series;  and  when  they 
decrease  by  a  common  divisor,  they  form  a  descending  geo- 
metrical series.  The  common  multiplier  or  divisor  is  called 
the  ratio. 

382.  In  an  ascending  geometrical  series,  each  succeeding 
term  is  found  by  multiplying  the  preceding  term  by  the  ratio. 

383.  The  following  symbols  are  used  in  geometrical  pro- 
gression, viz. :  a=  first  term,  /=  last  term,  n=  number  of 
terms,  r=  ratio,  and  S=  sum  of  the  terms. 

Using  the  above  symbols,  we  have  the  series, 
1    2     3     4.     5  71—4..  n— 3.  n— 2.  n—\.     n. 

a,  ar^  ar^^  ar^^  ar^  .  -  -  .  ar"^,  af*,  ar"^,  ar^-\  ar'^'K 
Hence  we  shall  have,* 

the   formula   for 


3.  Dividing,  evolving,  &c.,  r=  (-)**")  ( 


1 l=L       ,       ,       ^    , 

>      the  last  term. 

2.  Dividing,  &c.  .        .       fl=_L,        i  th«    f^^'""^^   f^' 
^  r*-'  >      the  first  term. 

the    formula   for 
the  ratio. 

These  three  formulas  may  be  enunciated  in  the  form  of 
general  propositions  or  rules. 

384.  If  the  series  is  descending,  the  above  formulas  may 
still  be  applied  by  taking  r=  to  the  reciprocal  of  the  com- 

•  The  formula  for  the  number  of  terms  is  solved  by  the  aid  of  logarithms, 
and  is,  consequently,  omitted  in  this  place. 


252  ELEMENTS    OF   ALGEBRA.  [SECT.  VIII. 

mon  divisor  ;  for,  multiplying  by  the  reciproca,!  of  a  number 
is  the  same  as  dividing  by  the  number  itself.  In  fact,  when- 
ever r<l,  the  series  will  be  descending. 

1 

385.  By  the  third  formula,  r=  [  -  j  "'~\  we  may  obtain  a 

general  method  for  finding  any  number  of  geometrical  means 
between  two  given  numbers.  To  do  this,  it  is  only  neces- 
sary to  obtain,  in  addition  to  the  given  data,  the  ratio. 

Let  m=  the  number  of  means ;  then,  since  the  whole  num- 
ber of  terms  in  the  series  consists  of  two  extremes,  plus  the 
means,  m-\-2=n. 


Henc( 


-G) 


When  the  ratio  is  found,  the  means  may  be  obtained  by 
the  continued  multiplication  of  the  first  extreme. 

PROBLEMS  TO  BE  SOLVED  BY  THE  PRECEDING 
FORMULAS. 

1.  The  first  term  of  a  geometrical  progression  is  5,  the 
ratio  4,  and  the  number  of  terms  7.     What  is  the  last  term  1 

Ans.  20480. 

2.  The  last  term  of  a  geometrical  series  is  98415,  the 
number  of  terms  11,  and  the  ratio  5.  What  is  the  first 
term  ^  Ans. 

3.  The  first  term  of  a  geometrical  series  is  28,  the  last 
term  20872,  and  the  number  of  terms  5,  What  is  the 
ratio  \  *^ns. 

4.  Find  two  geometrical  means  between  4  and  256. 

Jlns.  16  and  64. 

5.  The  first  term  of  a  geometrical  series  is  2,  the  number 
of  terms  8,  and  the  ratio  \,     What  is  the  last  term. 

^'^^'    8X92- 

6.  Find  three  geometrical  means  between  \  and  9. 

Jins.  i,  1,  and  3. 

7.  A  speculator  wishes  to  purchase  8  house  lots  of  a  land- 
holder, and  agrees  to  pay  for  the  8  lots  what  the  8  would 


SECT.  VIII.]  GEOMETRICAL    PROjGRESSION.  253 

come  to  if  the  first  be  valued  at  $2,  the  second  at  $6,  &c. 
What  did  he  pay  1  ^ns.  $4374.. 

8.  A  man  leased  a  plantation  on  condition  of  paying  for 
the  first  month  $1,  for  the  second  $2,  and  so  on  for  12 
i^nths.  At  the  end  of  10  months,  finding  he  had  made  a 
bad  bargain,  he  obtained  a  release  from  his  engagement  on 
condition  of  his  paying  what  would  have  been  the  stipulated 
sum  for  that  month.     How  much  did  he  pay  %    Ans,  $512. 

SUM  OF  THE  SERIES. 

386.  The  sum  of  the  series  may  evidently  be  obtained  by 
the  addition  of  all  the  terms,  but  it  is  necessary  to  obtain  a 
more  expeditious  method  for  finding  it. 

Using  thp  same  symbols»as  before,  we  have 

S=:a-}-ar+ar'-f-ar'+ f-ar^-^+ar'-'+ar''-'. 

And,rS=       ar-|-ar'+ar*-f-  ....  4-«r'*-''t-ar'-'+ar'»-'-|-ar\ 

Subtracting  the  first  equation  from  the  second, 

rS — S = ar" — a  ; 
Resolving  into  factors,  (r — l)S=ar" — a  ; 

Dividing         .         .        (1.)     Szzj^Z:?,  the  formula   for 

r —  1 

the  sum  of  the  terms  when  the  first  term,  the  number  of 

terms,  and  the  ratio  are  given. 

Or,  since  ar"—rxar"-^=rlj  i.  c,  the  last  term  multiplied 

by  the  r&tio, 

(2;)     S  = ,  the  formula  for  the  sum  of  the  terms  when 

the  first  term,  th^  last  term,  and  the  ratio  are  given. 

These  formulas  may  be  enunciated  in  the  form  of  general 
propositions  <or  rules,  and  applied  to  the  reduction  of  prob- 
lems. 

387.  From  the  above  formulas  it  appears  that  there  are 
five  things  to  be  considered  in  geometrical  progression,  viz. : 

1.  The  first  term. 

2.  The  last  terra. 

3.  The  ratio. 

22 


254  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII. 

4.  The  number  of  terms. 

5.  The  sum  of  the  terms. 

Any  three  of  these  being  given,  the  remaining  two^  except- 
ing the  number  of  terms,  may  be  found. 

388.  If  the  ratio  be  less  than  1,  the  progression  is  de- 
creasing ;  we  shall  also  have  r/<a. 

Hence  the  formula  for  the  sum  of  the  series  may  be  put 
under  the  form 

o_a — rl 
~  \—r' 

389.  To  obtain  a  formula  for  the  sum  of  the  terms  of  a 
decreasing  series  having  an  infinite  number  of  terms, 

Put  the  formula         -         S^r^H^", 

\-r.  ' 

Under  the  form  -         Sz=_iL— J^. 

Now,  since  r<l,  it  must  be  a  proper  fraction,  and  r"  is  a 
fraction  which  decreases  as  n  increases.     Therefore  will 

Xr"*  decrease   as  n  increases  j  and  when  n  be- 


1 — r     1 — r 

comes  greater  than  any  assignable  quantity,  or  when  n  be- 
comes infinite,  the  fraction  will  become  infinitely  small,  or 

=  0,  and  the  value  of  S  be  represented  by .    Hence 


1 — r  •  1 — r 

the  formula  for  the  sum  of  the  terms  of  a  decreasing  geo- 
metrical series,  in  which  the  number  of  terms  is  infinite,  is 


1— r  * 

This  is,  properly  speaking,  the  limit  of  the  decreasing 
series,  or  the  number  to  which  the  sum  of  the  terms  ap- 
proaches as  the  number  of  terms  increases  ;  but  it  can 
never  reach  this  number  until  an  infinite  number  of  terms  be 
taken. 

390.  The  above  formula  may  also  be  applied  to  the  sum- 
mation of  a  circulating  decimal  series,  as  3333,  &c.,  ad  in- 
fin,  ;  for  this  series  may  be  put  under  the  form  to+t^t4- 


SECT.  VIII.]  GEOMETRICAL    PROGRESSION.  255 

toVt+  iTT^ iToj  &C'     Hence,  the  first  term,  or  a  =  fV)  the  geo- 
metrical ratio,  or  r=-^,  and  the  sum 

&i=: =    ^    =1,  the  limit  of  the  series. 

2.  The  decimal  series,  323232,  &c.,  ad  injin.^  may  be  put 
under  the  form  -j?^«y-f-y^^^y,  &c.     Hence,  a=^^y  '•=TiT»  ^<i 

3.  The  decimal  series,  713333,  &c.,  ad  infin,,  may  be  put 
under  the  form  ^^~\-j^\^-^j^^^j^^  &c.    Hence,  TV?r  +  S=TVT 

1 — r  1 — j^ 

PROBLEMS  TO  BE  SOLVED  BY  THE  PRECEDING 
FORMULAS. 

1.  The  first  term  of  a  geometrical  series  is  1,  the  number 
of  terms  S,  and  the  ratio  5.  What  is  the  sum  of  all  the 
terms  1  ^ns.  97o56. 

2.  The  first  term  of  a  geometrical  series  is  6,  the  last 
1458,  and  the  ratio  3.     What  is  the  sum  of  all  the  terms  1 

^ns.  2184.. 

3.  The  last  term  of  a  geometrical  series  is  y^j,  the  ratio 
^,  and  the  sum  of  all  the  terms  7f||.  What  is  the  first 
term  %  ^ns.  4. 

4.  What  is  the  sum  of  the  series  1,  ^,  j,  &c.,  continued 
to  an  infinite  number  of  terms  1  ^ns.  2. 

5.  What  is  the  sum  of  the  series  1,  a,  ^,  &c.,  continued 
to  an  infinite  number  of  terms  1  *^ns.  \\, 

6.  The  first  term  of  a  geometrical  series  is  ^,  the  ratio  i, 
and  the  number  of  terms  5.  What  are  the  last  term  and 
the  sum  of  the  series  1  Jlns,*l=j\-^^  and  S=}f^. 

7.  The  first  term  of  a  geometrical  series  is  1,  the  ratio  |, 
and  the  number  of  terms  10.  What  is  the  sum  of  all  the 
term^l  ^ns.  VVoW- 

8.  A  person  being  asked  to  dispose  of  a  fine  horse,  said 
he  would  sell  him  on  condition  of  having  a  cent  for  the  first 
nail  in  his  shoes,  two  for  the  second,  four  for  the  third,  and 


256'  ELEMENTS    OF    ALGEBRA.  [sECT.  VIII. 

SO  on,  doubling  the  price  of  every  nail.  There  were  32  nails 
in  his  four  shoes.  What  would  the  horse  be  sold  for  at  that 
rate  1  ^ns,  $42949672,95. 

9.  A  man  failing  in  trade,  found  himself  in  debt  to  a  cer- 
tain amount,  after  he  had  given  up  all  his  property ;  but  his 
creditors  offered  to  employ  him,  giving  him  $1  for  the  first 
month's  service,  $3  for  the  second,  and  so  on  till  the  debt 
was  paid.  Having  accepted  the  offer,  he  found  that  it  re- 
quired of  him  but  10  months'  service  to.pay  the  debt.  What 
was  the  debt,  and  what  did  he  receive  for  his  last  month's 
services  1 

Ans.  Debt  $29,524,  and  he  received  for  his  last  month^s 

services  $19,683. 

10..  Two  couriers,  A  and  B,  set  out  at  the  same  time  to 
meet  each  other.  A  travels  6  miles  the  first  hour,  8  the 
second,  10  the  third,  and  so  on,  increasing  at  the  rate  of  2 
miles  every  hour.  B  goes  3  miles  the  first  hour,  4|  the  sec- 
ond, and  6f  the  third,  travelling  each  hour  1^  times  as  far 
as  the  preceding  hour.  They  meet  after  six  hours.  What 
is  the  distance  between  the  two  places  from  which  they  set 
out  1  Ans.  12811  miles. 

11.  Required  the  sum  of  the  decimal  series  ,81343434, 
&c.,  ad  injin.  .  Arts,  ffff . 

12.  Required  the  sura  of  the  three  following  series,  viz. : 

,111111,  &c.,  ad  injin. 
,232323,  (fee,  ad  injin. 
,714141,  &c.,  ad  injin.  Ans,  l|i^. 


• 


SECT.  IX.]  LOGARITHMS.  257 


SECTION  IX. 
Theory  of  Logarithms,  and  Construction  of  Logarithmic  Tables* 

391.  Logarithms  are  a  series  of  exponents,  computed  and 
arranged  into  tables  for  the  purpose  of  fa'cilitating  many  dif- 
ficult arithmetical  calculations. 

In  forming  a  system  of  logarithms,  some  number,  usually 
10,  is  selected  as  the  base  of  the  system.  Taking  10  as  the 
base  of  the  system,  then  the  logarithm  of  any  number  is  the 
exponent  denoting  the  power  to  which  10  must  be  involved 
to  produce  that  number. 

Let  a  represent  any  known  number,  and  x  the  unknown 
exponent  denoting  the  power  to  which  10  must  be  involved 
in  order  that  the  power  shall  equal  a  ;  we  shall  then  have 

10'=a. 

To  find  the  logarithm  of  a,  then,  requires  the  solution  of 
this  equation. 

392.  In  order  to  unfold  still  farther  the  theory  of  loga- 
rithms, and  a  method  by  which  the  logarithm  of  any  num- 
ber may  be  calculated,  let  us  take  a  geometrical  progression 
whose  first  term  is  unity  and#he  ratio  10  ;  and  also  an  arith- 
metical progression  whose  first  term  is  0,  and  whose  com- 
mon difference  is  unity. 

The  first  series  is  geometrical,^the  second  is  arithmetical. 

1,'  10,  100,  1000,  10000,  100000,  1000000,  10000000,  &c. 

0,    1,      2,        3,  4,  5,  6,  7,  &e. 

Supposing  the  two  series  to  be  continued  to  any  extent, 
the  numbers  in  the  arithmetical  series  are  called  the  loga- 
rithms of  the  corresponding  terms  in  the  geometrical  series  ; 
that  is,  they  arc  the  exponents  showitig  the  power  to  which 
10  must  be  involved  to  produce  the  corresponding  terms  in 
the  geometrical  series.  Thus  <Art.  107),  10°=  1,  10' =  10, 
10^=100,  10'=  1000,  &c. 

•  See  Note  D. 
Kk 


258  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

393.  From  the  nature  of  logarithms,  as  exhibited  above,  it 
will  be  easy  to  verify  the  truth  of  the  three  following  propo- 
sitions : 

1.  The  sum  of  the  logarithms  of  any  two  terms  of  the  geomet- 
rical series  is  the  logarithm  of  that  term  which  is  their  product. 

For  example,  the  sum  of  2  and  5,  the  logarithms  of  100 
and  100000,  is  7,  which  is  the  logarithm  of  10000000=100 
X 100000. 

2.  The  difference  of  the  logarithms  of  any  two  terms  of  the 
series  is  the  logarithm  of  thai  term  which  is  the  quotient  cf  the 
greater  divided  by  the  less. 

For  example,  the  difference  between  7  and  4,  the  loga- 
rithms of  10000000  and  10000,  is  3,  which  is  the  logarithm 
of  1000=  10000000^  10000. 

3.  The  arithmetical  mean  between  the  logarithms  of  any  two 
terms  in  the  series  is  the  logarithm  of  the  geometrical  mean  be- 
tween  those  terms. 

For  example,  ( 10  x  1000)^=  V  10000=  100,  which  is  the 
geometrical  mean  between  10  and  1000;  also,  (l-|-3)-7-2  = 
4<-i-2=2,  which  is  the  arithmetical  meajfi  between  the  loga- 
rithms of  10  and  100,  and  is  also  the  logarithm  of  100. 

394.  If  we  take  a  decreasing  geometrical  series  whose 
first  term  is  unity,  and  whose%atio  is  also  10,  we  shall  have 

11  1  1  1  _         •    1  i 1 Sirt* 

■*»     To"?      1  0  n>  To'o  0  J  I  0  0  0  o>  1  0  0  0  0  o>  1  0  0  0  0  0  OJ  1  0  0  0  0  0  0  o» '*''^*> 

Or,  1,  10-',  10-^  10-^      10-^,       10-^         10-^,  10-',  &c. 

Hence  the  corresponding  arithmetical  series,  or  the  loga- 
rithms, are 

0,     —1,     —2,     —3,     —4,     —5,     —6,     —7,  &c. 

395.  It  is  evident  that  the  logarithms  of  1,  10,  100,  &c., 
being  0,  1,  2,  &c.,  respectively,  the  logarithm  of  any  number 
between  1  and  10  will  be  0+  some  decirtial  parts  ;  that  of  a 
number  between  10  and  100,  1+  some  decinial  parts  j  that 
of  a  number  between  100  and  1000,  2+  some  decimal  parts, 
and  so  on  for  all  the  numbers  falling  between  the  successive 
terms  of  this  progression.- 

It  is  also  evident  that  the  logarithms  of  y\,  y^g^,  j^q-q^ 


SECT.  IX.]  LOGARITHMS.  259 

jjshz^  &c.,  or  .1,  .01,  .001,  .0001,  &c.,  being  —1,  ~2,  —3, 
— 4-,  &c.,the  logarithm  of  any  number  between  1  and  ,1  will 
be  — 1-h  some  positive  decimal  parts  ;  that  of  a  number  be- 
tween .1  and  .01,  — 2-f-  some  decimal  parts;  that  of  a  num- 
ber between  .01  and  .001,  — 3-|-  some  decimal  parts,  &c. 

396.  Between  each  two  adjoining  terms  of  both  series  in 
Art.  392  a  term  may  be  interpolated,  and  a  new  series  of 
numbers, and  logarithms  will  be  produced,  each  consisting 
of  double  the  number  of  terms.  T4iis  interpolation  nruiy  be 
effected  by  finding  the  geometrical  mean  (Art.  365,  Cor.),  or 
taking  the  square  root  of  the  product  of  the  two  terms  in 
the  geometrical  series,  and  the  arithmetical  mean^  or  half 
their  sum,  in  the  arithmetical  series.  The  term  interpola- 
ted between  1  and  iO  in  the  geometrical  series  would  be 
>/lX  10=3,1622777;  between  10  ©nd  100  would  be  v/lOx" 
100=31,62280  ;  the  corresponding  terms  interpolated  in  the 
arithmetical  series  would  be  (0+l)-r2=,5,  and  (l-h2)-r2 
=  1,5.       ' 

The  two  series,  then,  would  be 

1,     3.162277,     10,     31.62280,     100,  &c. 

0,  .5,       1,  1.5,         2,  &c. 

These  two  series  may  again  be  interpolated  as  before,  and 

so  on  continiKilly.     The  nulhber  of  terms  in  the  two  series 

will  continually  increase,  and  the  differences  between  them 

continually  decrease,  with  each  succeeding  interpolation* 

397.  To  construct  a  table  of  logarithms,  however,  it  is 
unnecessary  to  interpolate  systematically  throughout  the 
series  ;  for,  if  the  logarithm  of  some  few  of  the  prime  num- 
bers be  calculated,  those  of  the  composite  numbers  may  be 
obtained  by  the  process  indicated  in  Art.  393.  Indeed,  these 
interpolations  may  be  limited  to  any  two  adjoining  terms  in 
the  series. 

Hence,  the  logarithm  of  any  number,  whele  or  fractional, 
between  any  two  terms  of  the  series  in  Art.  392,  may  be  cal- 
culated by  the  following  general 


260  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

RULE* 

1.  Find  a  geometrical  mean  between  1  and  10,  10  and  100, 
or  any  other  two  adjacent  terms  of  the  series  between  which  the 
number  proposed  lies.  Also^  between  the  mean  thus  found  and 
the  nearest  extreme^  observing  that  the  proposed  number  shall  fall 
between  the  mean  found  and  that  extreme^  find  another  geometri- 
cal mean^  as  before  ;  and  so  on,  till  you  have  arrived  sufficiently 
near  the  number  whose  logarithm  is  sought. 

2.  Find  as  many  arithmetical  means  between  the  correspond, 
ing  terms  of  the  arithmetical  series  0,  1,  2,  3,  <^c.,  in  the  same 
order  as  the  geometrical  means  were  found,  and  the  last  of  these 
will  be  the  logarithm  of  the  proposed  number 

EXAMPLES.     • 

1.  Calculate  the  logarithm  of  5. 

Here  the  proposed  number  lies  between  1  and  10. 

Firsty  then,  the  logarithm  of  10  is  1,  and  the  logarithm  of 
1  is  0. 

Then,  (10  X  1)^  =  3.162277,  which  is  the  geometrical  mean, 

And,  (l  +  0)-r-2— 1  =  .5,  which  is  the  arithmetical  mean. 

Hence,  the  logarithm  of  3.162277  is  .5. 

Secondly,  the  logarithm  of  10  is  1,  an<i  the  logarithm  of 
3.162277  is  ,5. 

Then,  (3.162277  x  10)*=:5.623413,  which  is  the  geometric 
cal  mean, 

And  -  (l-f, 5) -7-2=0.75,  which  is  the  arithmetical 
mean. 

Hence,  the  logarithm  of  5.623413  is  0.75. 

Thirdly,  the  logarithm  of  5.623413  is  0.75,  and  the  loga~ 
rithmof  3.162277is  0.5. 

Then,  (5.623413  x  3.162277)^  =:4.216964,  which  is  the  ge- 
ometrical mean,. 

And  -  -  (0.54-0.75)-f-2=0.625,  which  is  the  arith- 
metical mean. 

Hence,  the  logarithm  of  4.216964  is  0.625. 

Fourthly,  the  logarithm  of  5.623413  is  0.75,  and  the  loga- 
«ithm  of  4-216964  is  0.625. 


SECT.   IX.] 


LOGARITHMS. 


261 


Xhen,  (5.623413  x4.216964)^=4..869674,  which  is  the  ge- 
ometrical mean, 
And       -       (0.75+0.625)^2=0.6875,  which  is  the  arith- 
metical mean. 
Hence,  the  logarithm  of  4.869674  is  0.6875. 
Fifthly,  the  logarithm  of  5.623413  is  0.75,  and  the  loga- 
rithm of  4.869674  is  0.6875. 

Then,  (5.623413  x  4.869674)^=5.232991,  which  is  the  geo- 
metrical mean, 
And       -     (0.75-h0.6875)-7-2=0.71875,  which  is  the  arith- 
metical mean. 
Hence,  the  logarithm  of  5.232991  is  0.71875. 
Proceeding  in  this  way,  the  22d  geometrical  mean  will  be 
found  to  agree  with  5,  as  far,  at  least,  as  the  sixth  place  of 
decimals ;  hence,  for  all  practical  purposes,  they  may  be  con- 
sidered equal,  and  the  22d  term  in  the  corresponding  arith- 
metical series  be  taken  as  the  logarithm  of  5. 

These  operations,  and  their  results,  may  be  expressed  in 
the  following  table  : 

Numben. 

1.  (10x1)*  =3.162277, 

2.  (10x3.162277)*  =5.623413, 

3.  (3.162277  x5.623413)*=4.216964, 


(5.623413  X  4.216964)*  =4.869674, 
(5.6234 13  X  4.869674)*  =  5.23299 1, 
(4.869674  x  5.23299 1;*  =  5.048065, 
(4.869674  x  5.048065)* =4.958069, 
(5.048065  X  4.958069)*  =  5.002865, 
(4.958069  X  5.002865)*=4.9804.16, 
(5.002865  x4.980416)*=4.991627, 
(5.002865  x4.991627)*=4.997240, 

12.  (5.002865  x  4.997240)*  =  5.000052, 

13.  (4.997240  x  5.000052)* =4.998647, 

14.  (5.000052  X  4.998647)* =4.999350, 


4. 
5. 
6. 

7. 

8. 

9. 
10. 
11. 


Lnpu-itbmf. 

0.50000000. 


0.75000000. 
0.62500000, 
0.68750000. 
0.71875000. 
0.70312500. 
0.69531250. 
0.69921875, 
0.69726562. 
0.69824218. 
0.69873046. 
0.69897460. 
0.69885254r. 
0.69891357. 


ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

Numben.  Logarithms. 

15.  (5.000052x4.999350)^=4.999701,  0.69894409. 

16.  (5.000052x4.999701)^  =.4.999876,  0.69895935. 

17.  (5.000052  X  4.999876)^=z4.999963,  0.6989668. 

18.  (5.000052  X  4.999963)^=5.000008,  0.6989707. 

19.  (4.999963  x  5.000008)^=4.999984,  0.6989687. 
^0.  (5.000008  X  4.999984)^=4.999997,  0.6989697. 

21.  (5.000008x4.999997)^  =  5.000003,    0.6989702. 

22.  (4.999997x5.000003)^-5-000000,         0.6989700. 
J\^ote  1. — A  greater  degree  of  exactness  might  be  attained 

by  carrying  out  the  work  to  a  greatpr  number  of  decimal 
places,  and  continuing  our  interpolations. 

J^ote  2. — Having  thus  obtained  the  logarithm  of  5,  and 
that  of  10  being  given,  the  logarithm  of  2  can  be  readily 
found  ;  for,  since  10-r5  =  2,  logarithm  of  10,  minus  loga- 
rithm of  5=  logarithm  of  2,  or  1—0.6989700=0.3010300, 
which  is  the  logarithm  of  2. 

2.  Required  the  logarithm  of  3.  Jlns.  0.47712125. 

3.  Required  the  logarithm  of  7.  .^ns.  0.84509804. 

398.  The  great  difficulty  of  constructing  a  table  of  loga- 
rithms is  in  finding  the  logarithms  of  the  prime  numbers. 
These  were  first  computed  by  successive  interpolations,  as 
in  the  preceding  examples.  The  logarithms  of  composite 
numbers  are  found  by  adding  the  logarithms  of  the  factors 
whose  product  is  equal  to  the  composite  number. 

399.  The  computation  of  the  logarithms  of  prime  num- 
bers, after  the  logarithm  of  2  has  been  obtained,  may  be 
greatly  abridged  by  the  following  general 

RULE.* 

When  the  logarithm  of  any  number  (n)  is  known,  the  log- 
arithm of  the  next  greater  number  may  be  readily  found  by 
substituting  the  numerical  value  of  the  letters  in  the  follow- 
ing series,  and  then  calculating  a  sufficient  number  of  terms. 

Let  n=  the  jiumber  whose  logarithm  is  given,  ?i-|-l=  the 
*  See  Note  E. 


SECT.  IX.] 


LOGARITHMS. 


263 


number  whose  logarithm  is  to  be  found,  and  M=  the  modu- 
lus of  the  system  =0.4342944.819,  or  2M=0.8685889638. 
Then  will 

Lo?arit)im  (»-f  1)=  logarithm  n-f-2M( +7r^r r^ 

5(2;i+  ly     7(2;i-}- 1)'     9(2/1 -h  1)7 
Or,  letting  A,  B,  C,  D,  dec,  represent  the  terms  immedi- 
ately preceding  those  in  which  they  are  used, 

2M  A 

Logarithm    (71+ 1)=    logarithm    n  +  ^_  ^  i^-^th — r^n"^ 


&c. 


SB 


JI+; 


logarithm 
5C  7D 


2n-f-l 


3(271+1)^ 


5(271-1-1)*     7(271 -hi)'     9(27»-hl)' 

EXAMPLES. 

1.  Required  the  logarithm  of  3. 

Here,  since  72-|-l  =  3,  7i  =  2,  and  27i-f-l  =  5,  we  shall  have 


Logarithm  n 

2M 

2/i4  1 

A 


3(271 -hi)* 

3B 
5(271  +  1)* 

5C 

7(2«4-l)' 

7D 
9(2/t-Kl)  = 
9E 

11(271 -hi)* 


rlogarithm  2 
0.868588964 

5 
0.173717793 

3>r5^ 
3x0.002316237 

5x0.000055590 


7x5' 
7x0.000001588 
''^     9x5' 
9x0.000000050 


=0.301029995 
=0.173717793 

=0.002316237 

=0.000055590 

=0.000001588 

=0.000000050 

=0.000000002 


(A.) 
(B.) 
(C.) 
(D.) 
(E.) 
(F.) 


11x5*  

Whence  the  log.  (2-hl)=  log.  3      =0.477121255. 

The  above  logarithm  is  correct  as  far  as  to  the  ninth  place 
of  decimals. 

2.  Required  the  logarithm  of  11.  ^ns.  1.04139269. 

400.  The  only  numbers  whose  logarithms  it  will  be  found 
necessary  to  compute  by  the  preceding  formula,  of  by  in- 
terpolating the  series,  are  the  prime  numbers  3,  7,  11,  13, 


264  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

17,  19,  23,  29,  &;c.     The  logarithms  of  composite  numbers 
may  be  computed  by  the  propositions  verified  in  Art.  393. 

COMPUTATION  OF  LOGARITHMIC  TABLES. 
401.  The  following  table  will  exhibit  the  manner  in  which 
the  logarithms  of  the  natural  series  of  numbers  1,  2,  3,  4, 
&c.,  to  30,  may  bccomputed:  '  4 

Nos.  Method  of  Computation.  Logarithmi. 

1. log.  Izz:  0.00000000. 

2.  Since  10H-5  =  2,log.  10- log.  5=  log.  2=0.30103000. 

3.  Computed  by  formula  in  Art.  399,   log.  3  =  0.47712126. 

4.  Since  2x2=4,  log.  2+  log.  2    =  log.  4  =  0.60206000. 

5.  Computed  by  interpolating  the  >      j  5=0  69897000. 

series  in  Art.  397  -  5  '      ^' 

6.  Since  2x3  =  6,  log.  27f  log.  3    =  log.  6=0.77815125. 

7.  Computed  by  formula  in  Art.  399,    log.  7  =  0.84509804. 

8.  Since  2X4=8,  log.  2+ log.  4    =  log!  8  =  0.90308999. 

9.  Since  3x3=9,  log.  3+ log.  3     =  log.  9  =  0.95424251. 

10.  :-..-.       log.  10=1.00000000. 

11.  Computed  by  formula  in  Art.  399,   log.  11  =  1.04139269. 

12.  Since  3x4=12,  log.  3+ log.  4  =  log.  12=1.07918125. 

13.  Computed  by  formula  in  Art.  396,   log.  13=1.11394335. 

14.  Since2x7=14,  log.  2+log.7   =  log.  14=1.14612804. 

15.  Since3x5  =  15,log.  3+log.5   =  log.  15  =  1.17609126. 

16.  Since  4x4=16,  log.  4+ log. 4   =  log.  16=1.20411998. 

17.  Computed  by  formula  in  Art.  399     log  17=  1.23044892. 

18.  Since3x6  =  18,  log.  3+log.6   =  log.  18=1.25527251. 

19.  Computed  by  formula  in  Art.  399,  log.  19  =  1.^7875360. 

20.  Since  2  x  10=20,  log.  2+  log.  10=  log.  20=  1.30103000. 

21.  Since  3x   7=21, log. 3+ log.   7=  log.^l  =  1.32221929. 

22.  Since  2x  11  =  22, log. 2+ log.  lI=log.  22=  L34242268. 

23.  Computed  by  formula  in  Art.  399,   log.  23=1.36172784. 

24.  Since4x    6  =  24, log.4+ log.    6=  log. 24=  1.38021124. 

25.  Since5x    5=25, log. 5.+ log.    5=  log. 25  =  1.39794001. 

26.  Since  2  x  13  =  26,  log.  2+  log.  13=  log.  26  =  1.41497335. 
27.- Since  ^X  9  =  27,  log.  3 -f- log.  9=  log.  27=  1.43136376. 
28.  Since  4  X   7=28,  log.  4+ log.  7=  log.  28=  1.44715803. 


SECT.  IX.]  LOGARITHMS.  265 


29.  Computed  by  formula  in  Art.  399,  log.  29  =  1.46239800. 

30.  Since  3  X  10=30,  log. 3-Mog.4.  =  log.  30=  1.47712125. 
The  logarithm  usually  consists  of  two  parts,  the  integral 

part,  usually  called  the  index  or  characteristic^  and  a  decimal. 

402.  It  will  also  be  perceived  that  the  multiplying  or  di- 
viding of  any  number  by  10,  100,  1000,  &c  ,  is  performed  by 
increasing  or  diminishing  the  integral  part  of  its  logarithm 
by  1,  2,  3,  &c.  J  hence,  all  numbers  which  consist  of  the 
same  figures,  whether,  they  be  integers,  decimals,  or  mixed 
numbers,  will  have  for  the  decimal  part  of  their  logarithms 
the  same  positive  number. 

Thus,  according  to  the  tables  now  in  common  use,  the 
logarithm  of  3854  is  3.58591171. 
Log.  3854  =3.58591171  J 

Log.  38540=  log.  (3854  x  10)=  log. 3854+ 1=4.58591171  j 

Log.  385,4  =  log.  ??^  =  log.  3854-1=2.58591171 ; 

Log.  38,54  =  log.^^  =  log. 3854-2=1.58591171  J 

Log.  3,854  =  log.  ??5^  =  log.  3854-3=0.58591 171 ; 

Log.  ,3854  =  log.  ^?5i        =  log.  3854-4=  1.58591171 ; 
^  '  ^   10000  ^ 

Log.  ,03854=  log.  J?5i.      =  log.  3854-5=2.58591171. 
^  ^  100000  ^ 

The  number  of  units  in  the  characteristic  of  a  logarithm 

is  one  less  than  the  number  of  digits  in  the  natural  number  j 

and  for  decimals,  the  negative  characteristic  denotes  how 

far  the  first  significant  figure  is  removed  from  the  place  of 

units.     The  decimal  part  of  the  logarithm  is  always  positive. 

APPLICATIONS  OF  LOGARITHMS. 

403.  The  tables  of  logarithms  in  common  use  contain  the 
logarithms  of  numbers  from  1  to  10000.  An  explanation  of 
these  tables,  and  also  of  the  methods  of  finding  from  them 
the  logarithm  of  any  number,  or  the  number  of  any  logarithm 

23  L  L 


266  ELEMENTS    OF   ALGEBRA.  [sECT.  IX. 

whatever,  usually  accompanies  them,  so  that  such  explana- 
tions are  unnecessary  here.  The  numbers  and  logarithms 
used  in  the  following  applications  of  logarithms  are  taken 
from  these  tables. 

I.    MULTIPLICATION    AND    DIVISION. 

404.  Since  logarithms  are  a  series  of  exponents  denoting 
different  powers  of  the  common  number  10,  it  is  evident 
that  the  sum  of  the  logarithms  of  any  two  numbers  will  be 
the  logarithm  of  their  product,  and  the  difference  of  their 
logarithms  will  be  the  logarithm  of  the  quotient  produced 
by  dividing  the  greater  by  the  less.     Hence, 

I.  To  multiply  by  logarithms,  take  the  logarithms  of  the  fac- 
tors from  the  table^  add  them  together^  and  then  find  the  natural 
number  corresponding  to  their  sum  ;  this  will  he  the  product  re- 
quired. 

1.  Multiply  16  by  5,  by  logarithms. 

Logarithm  16  =1.20411998; 

Logarithm    "5  =z0.69897000 ; 


Logarithm  80  =1.90308998.       Ans.QO, 

2.  Multiply  37153  by  4086,  by  logarithms. 

Logarithm  37153       =4.5699939  ; 
Logarithm  408,6        =2.6112984; 

Product,  15180715.8  .  .  6.1812923. 

3.  Muhiply  4675,12  by  .03275,  by  logarithms. 

Logarithm  4675.12    =3.6697928; 
Logarithm  0  03275    =2.5152113; 

Product,  153,1102,  &c.  .  2.3850041. 
II.  To  divide  by  logarithms,  subtract  the  logarithm  of  the 
divisor  from  the  logarithm  of  the  dividend,  and  the  remainder 
will  be  the  logarithm  of  the  quotient. 

EXAMPLES. 

1.  Divide  72  by  24,  by  logarithms. 

Logarithm  72  =1.85733250; 

Logarithm  24  =1.38021124; 

Quotient,  3 0.47712126. 


•BCT.  IX.]  LOGARFTHMS.  26T 

2.  Divide  4768,2  by  36,954,  by  logarithms. 

Logarithm  4768,2      =  3  6783545  ; 

Logarithm  36,954      =1  5676615; 

Quotient,  129,032  ....  2.1106930. 

3.  Divide  46257  by  ,17608,  by  logarithms. 

Logarithm  46257      =4.6651725; 

Logarithm  ,17608      =1.2457100; 

Quotient,  2^2741  .  .  .  .  5.4194625.- 

II.    INVOLUTION  AND  EVOLUTION. 

405.  Involution  is  performed  by  multiplying  the  exponent 
of  the  number  to  be  involved  by  the  exponent  denoting  the 
power  (Art.  157) ;  and  Evolution  by  dividing  the  exponent 
of  the  number  by  the  exponent  denoting  the  root  to  be  ta- 
ken (Art.  289).     Hence, 

I.  To  involve  by  logarithms,  multiply  the  logarithm  of  the 
number  to  be  involved  by  the  number  denoting  the  power  ;  the 
product  will  be  the  logarithm  of  the  power. 

EXAMPLES. 

1.  Involve  9  to  the  second  power,  by  logarithms. 

Logarithm  9           =0.95424251; 
Multiplying  by  2,  2  ; 

Square,  81 1.90848502. 

2.  Involve  7.0851  to  the  third  power,  by  logarithms. 

Logarithm  7.0851  =0.8503399 ; 

Multiplying  by  3,  3  ; 

Cube,  355,6475  ....  2.5510197. 

3.  Involve  0.9061  to  the  seventh  power,  by  logarithms. 

Logarithm  0.9061  =1.9571761304  ; 

Multiplyihg  by  7,       7  ; 

Power,  0.5015  ....  1.7002329128. 

4.  Involve  1.0045  to  the  365th  power,  by  logarithms 

Logarithm  1.0045  =0.0019499  ; 

Multiplying  by  365,  365  ; 

97495" 

116994 

58497 


Power,  5.148888  .  .  .  0.7117135. 


268  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

II.  To  evolve  by  logarithms,  divide  the  logarithm  of  the 
given  number  by  the  number  denoting  the  root  to  be  taken  ;  the 
quotient  will  be  the  logarithm  of  the  root. 

EXAMPLES. 

1.  Evolve  81  to  the  fourth  root,  by  logarithms. 

Logarithm  81  =r  1.90848502  ; 

Dividing  by  4,  -r-4  ; 

Root,  3 0.47712120. 

2.  Evolve  7.0825  to  the  fifth  root,  by  logarithms. 

Logarithm  7.0825  =0.8501866  j 
Dividing  by  5,  4-5  j 

Root,  1.479235  ....  0.1700373. 

3.  Evolve  1.045  to  the  365th  root,  by  logarithms. 

Logarithm  1.045    =0.0019116; 
Dividing  by  365,  -i-365j 

Root,  1.000121   ....  0.0000524. 

4.  What  is  the  8th  power  of  the  9th  root  of  654  % 

Logarithm  654       r=2.8 155777483  ; 
Multiplying  by  8,  8 ; 

*  22.5246219864 ; 

Dividing  by  9,  -r-9  j 

Root,  318.3 2.507357762. 

III.    EXPONENTIAL  EQUATIONS. 

406.  Equations  into  which  the  unknown  quantity  enters 
in  the  form  of  an  index  are  called  exponential  equations. 

Such  equations  may  be  most  readily  solved  by  logarithms. 
Thus,  d^=b^  but  a=^=(log.  a)xx ;  therefore,  (log.  a)xx= 

log.  5,  or,  dividing  by  log.  a,  x—  ^^'    . 

log.  a 

EXAMPLES. 

1.  Reduce  the  equation  5''i=100. 

As  the  two  members  are  equal,  their  logarithms  must  also 
be  equal  y  therefore, 


SECT.  IX.]  LOGARITHMS.  269 

(Log.  5)xa?=  log.  100; 

Ti'     .'                                 log.  100     2  00000000    OQC1 
Dividing      -        -      x=  _-° = — — =2.861. 

log.  5       0.69897000 

2.  Reduce  the  equation  3'=24.3. 

(Log.  3)xx=  log.  24.3; 

Tx-  ...  log.  243     2.38561     ^ 

Dividing       -         -      x=  —P- = =5. 

^  log.  3       0.47712 

407.  Another  and  a  more  difficult  form  of  exponential 

equation  is  a"*=b.     Here  the  exponent  x  is  the  exponent 
of  the  exponent  m. 

In  this  equation  assume  rrfzuy^  then  a?=by 

And     -         (log.  a)xy=  log.  h  ; 

Dividing       -         -       y=  ^^'     ; 
log.  a 

Hence  -        -    »t*= —°^  (which  let) =c; 

log.  a 

Then   -       (log.  »i)xa?=  log.  c; 

Dividing      •        -      ir=    ^^'  ^ . 
log.  m 

EXAMPLE. 

X 

1.  Reduce  the  equation  9'=  1000. 

(Log.  9)  X  3'=  log.  1000  J 
Dividing      .        .     3x^  log.  1000^3.00000000^3^ 

^  log.  9        0.95424251  ' 

Then   -        -        -     3*=  3.14.-.  (log.  3) Xic=  log. 3.14, 

And     -        .         .       x=l^^Jll!=2^9^5-1.04. 

log.  3       0.47712126 

2.  Reduce  the  equation  4^=4096. 

^n«.a:=  1:^=1.6309  +  . 
log.  3 

408.  A  third  and  a  still  more  difficult  form  of  the  expo- 
nential equation  is  af=b. 

Taking  the  logarithms  of  both  sides,  we  have 

(Log.  x)xx=  log.  b. 
This  equation  naay  be  solved  by  "  Trial  and  Errors    Thus, 


270  ELEMENTS    OF   ALGEBRA.  [sECT.  IX. 

make  two  suppositions  of  the  value  of  the  unknown  quanti- 
ty, and  find  their  errors  j  then  institute  the  following  pro- 
portion :  / 
Diff.  of  the  errors  :  diff.  of  the  assumed  numbers  :  :  least 

error  :  to  the  correction  required  in  the  corresponding 

assumed  number. 

EXAMPLES. 

1.  Reduce  the  equation  a;*=i256. 

Then     -     (log.  x)xcc=  log.  256  ; 
Suppose         -         -    07=3.5,  or  3.6.  ' 

By  first  Supposition.  By  second  Supposition. 


Log.  x=[og,  3.5=0.54406804 
Multiplying  by  3  5 


(Log.  a?)xa7=  1.90423814 
But,  log.  256      =2  40823997 


Error*        -         0.50400183 


Log.  X-  log.  3.6=0.55630250 
Muhiplying  by  3.6 

(Log.  x)xx        =2.00263900 
Log.  256  =2.40823997 

Error*        -        0.39555097 


Difference  of  the  errors,  0.10844086. 
Then,  0.10844086  :  0.1  :  ;  0.39555097  :  0.365  j  hence  x 

3.965  +  . 
To  correct  this  still  farther,  suppose  a?=3.96,  or  4.01. 

By  first  Supposition.  By  second  Supposition. 

Lo^.a;=loa.  3.96=0.59769519 


Multiplying  by  3  96 


(Log.  a;)xa:         =2  36687295 


Loa.x=  log.  4,01=0.60314437 
Multiplying  by  4.01 

{Log.  x)  XX         =2.41860892 


256  =2.40823997,Lo^.  256  =2.40823997 


First  errorf     =0.04136702!     Second  errort=0.01036895 

Difference  of  the  errors  =0  05173597. 
Then,  0  05173597  :  0.05  : :  0.01036895  :  0.01. 
Hence,  a:=4.01  — 0.01=4,  which  value  for  x  satisfies  the 
conditions  of  the  equation  ;  for 

4^=256. 
2.  Reduce  the  equation  4a;''=100a?^ 

*  Both  these  suppositions  are  discovered  to  be  less  than  the  true  number; 
hence  the  errors  are  like  with  reference  to  their  signs. 

t  One  of  these  suppositions  is  less,  the  other  greater  than  the  true  value 
of  a;;  hence  the  errors  are  unlike  with  reference  to  their  signs. 


•ECT.  IX.] 


LOGARITHMS. 


271 


Dividing  by  4       -         -         -         -      af=25a^; 
Dividing  by  a:*      ....    jf-«=r25j 
Taking  the  logarithm,  (log.  af)x(x— 3)=  log.  25; 
Suppose x=4fy  or  6. 

Bjr  wcoDd  Suppokition. 

Log.  x=  log.  6   =0.77815125 


By  fitrt  SappMition. 

Log.  x=  log.  4    =0.60205999 
Multiplying   }     _  , 


ultiplying   ) 
by  X— 3      S 


(Log.  x)  X  (ar— 3)  =p  0.60205999 
Log.  25  =1.39794001 


First  error       =0.79588002 

Difference  of  the  errors  =1.73239376. 
Then,  1.73239376  :  2  : :  0.79588002  :  0.092 
Hence,  x=4-|-0.92=4.92. 
Again,  suppose  07=4.92,  or  4.93. 

By  Am  Supptwitioo. 

Log.x=log.4.92=0.69196510 
Multiplying   ^     ^  J  ^2 

:— 3      S 


Multiplying^ 

by  X — 3 
(Log.  a)  X  (a:— 3) =2.33445375 
Log.  25       =1.39794001 

Second  error  =0.93651374 


by  X- 
(Log.  x)  X  (x--3)= 1 .32857299 


By  tecood  Sappotilion. 

Log.  x= log.  4.93  =  0.69284692 
Multiplying 
by  X— 3 


!  = 


1.93 


(Log.  x)  X  (X— 3)  =  1 .337 1 9456 


1.39794001'Log.  25 


=  1.39794001 


Log.  25 

First  error       =0.06936702,'     Second  error  =0.06074545 

Difference  of  errors  =0.00851157. 
Then,  0.00851157  :  0.01  : :  0.06074545  :  0.07. 
Hence,  i= 4. 93 -1-0.07  — 5.00,  which  value  for  x  satisfies 
the  conditions  of  the  equation  ;  for 

4X5^=100X5'. 

IV.    GEOMETRICAL  SERIES. 

409.  Logarithms  are  also  very  convenient  in  finding  the 
last  term,  and  also  the  sum  of  the  series  in  Geometrical 
Progression,  when  n  is  not  a  very  small  number.  The  num- 
ber of  terms  may  also  be  obtained  by  the  aid  of  logarithms. 

I.  The  formula  for  the  last  term  is  (Art.  383), 

Taking  the  log.,     log.  /=  log.  a4-(log.  r)x(n  —  \). 
Hence,  to  find  the  last  term  in  n  geometrical  series  by 


272  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

logarithms,  add  the  logarithm  of  the  first  term  to  the  logarithm 
of  the  ratio  multiplied  by  the  number  of  terms  less  one  ;  the  sum 
will  be  the  logarithm  of  the  last  term, 

EXAMPLE. 

1.  The  first  term  of  a  geometrical  series  is  4,  the  ratio  5, 
and  the  number  of  terms  61.     Required  the  last  term. 

Or,  log.  l=z  log.  4+(log.  5)  X  60=0.60205999+41.9382000 

=  42.54025999.  ' 

Hence,  finding  the  natural  number  corresponding  with 
42.54025999, 

7=3469479392577934009746744427570344331708876. 

2.  The  formula  for  the  sum  of  the  terms  in  a  geometrical 
series  is  (Art.  386) 

g_ar"— <x         ^ 
r— 1  * 
In  this  formula,  if  n  is  not  a  small  number,  it  will  he  found 
convenient  to  find  the  value  of  ar"  by  taking 
Log.  (ar'^)=  log.  <z-l-(log.  r)xn. 
Thus,  we  may  find  the  value  of  ar""  in  the  same  way  that 
we  found  the  last  term  in  the  preceding  case, 

3.  To  obtain  a  formula  for  the  number  of  terms,  let  us 
resume  the  formula  for  the  sum  of  the  terms, 

b= .— ; 

Clearing  of  fractions,      rS— S=ar" — a; 
Transposing         -         -         ar'*=rS  — S+a; 


Dividing 


a 


Hence,         -         -    (log.  r)X7i=  log.  (rS— S+a)— log.  a/ 

D.   •!•                                                 log.  (rS— S4-«)— loor.  a 
ividmg      -        -        -        -  n— — ^— ^ —L 2 , 

log.  r 

EXAMPLES. 

1.  The  sum  of  a  geometrical  series  is  6560,  the  first  term 
2,  and  the  ratio  3.     Required  the  number  of  terms. 
Here-         -       S=6560,  a=2,  and  r=3  j 


SECT.  IX.]  LOGARITHMS.  S79 

Hence        -        n=}2iL0±±t^t±Si^', 

log.  r 

—  l£gl_l?122~Jlog^2 . 
~  log.  3 

_  3.8169700_Q 
""0.4771213" 

2.  The  sum  of  a  geometrical  series  is  1023,  the  first  term 
1,  and  the  ratio  2.     Required  the  number  of  terms. 

jJns.  10. 

3.  The  sum  of  a  geometrical  series  is  640,  the  first  term 
4,  and  the  ratio  1.01.     Required  the  number  of  terms. 

V.    COMPOUND  INTEREST. 

410.  By  means  of  logarithms  we  may  also  determine  the 
number  of  years  it  will  take  a  given  principal,  at  a  given 
rate,  compound  interest,  to  gain  a  certain  amount.  Thus,  in 
Art.  219,  we  have  the  formula 

A=P(l+r)"; 
Taking  the  logarithms,  log.  A=  log.  P+(log.  (l+r))x»  .• 
Transposing,  (log.  {l-\-r))xn=  log.  A—  log.  P; 

_  log.  A—  log.  P 
log.  (1+r) 
J^ote. — By  means  of  this  formula  we  may  ascertain  the 
number  of  years  it  would  take  a  sum  of  money  to  double, 
triple,  &c.,  or  amount  to  m  times  itself,  when  put  out  at 
compound  interest,  at  a  given  rate  per  cent. 

EXAMPLES. 

1.  A  man  loans  $1250,  at  6  per  cent,  compound  interest. 
In  what  time  will  it  amount  to  $4008.92  1 

In  this  example,  A=$4008.92,  P=1250,  and  r+l=,06-f- 
1  =  1.06. 

Hence  n-  ^^^'  ^^^^■^^-  ^^g-  ^^^^  . 

_3.60302739-3.09691001^^^  ^^^. 
0.02530587  ^* 

2.  At  6  per  cent,  compound  interest,  in  how  many  yean 
will  $1200  amount  to  $2149.191  ^ns,  10  years. 

Mm 


274  ELEMENTS    OF    ALGEBRA.  [sECT.  IX. 

3.  At  6  per  cent,  compound' interest,  in  how  many  years 
will  money  amount  to  double,  triple,  and  quadruple  the  ori- 
ginal sural  Ans.  11,9955,  18,8145,  and  23,791  years. 


APPENDIX. 


The  three  following  sections,  as  they  contain  those  portions  of 
algebraic  analysis  which  are  seldom  pursued  in  academies  and 
schools,  but  which  are  nevertheless  essential  to  the  successful 
prosecution  of  the  higher  branches  of  the  mathematical  course, 
have  been  thrown  into  the  form  of  an  appendix. 


APPENDIX. 


SECTION  X. 

Permutations^  ,^rrangementSy  and  Combinations. — DemonstrO' 
lion  of  the  Binomial  Theorem. — Continued  Fractions. — /n- 
finite  Series. — Expansion  of  Infinite  Series. — Indeterminate 
Coefficients. — Summation  of  Infinite  Series. — Recurring  Se- 
ries.— Method  of  Differences. — Reversion  of  Series. 

PERMUTATIONS,  ARRANGEMENTS,  AND  COMBINATIONS. 

I.    PERMUTATIONS. 

411.  Permutations  are  the  results  obtained  by  writing  a 
given  number  of  letters,  one  after  the  other,  in  every  possi- 
ble way,  in  such  a  manner  that  all  the  letters  may  enter  into 
each  result,  and  each  letter  enter  but  once. 

Thus,  the  two  letters  a  and  b  furnish  the  two  per-  i  ab 
mutations t  ba 

The  three  letters,  a,  J,  and  c,  furnish  six  permutations,  viz. : 


abcj  acby  cah^  cba^  bac,  bca. 

Hence,  the  permutations  of  three  letters  are  equal  to  the  per- 
mutations of  two  letters  multiplied  by  three. 

412.  In  like  manner,  the  permutations  of  four  letters  will  be 
found  equal  to  the  permutations  of  three  letters  multiplied  by 
four. 

And,  in  general,  the  permutations  of  any  number  whatever 
(n)  of  letters  will  be  equal  to  the  permutations  of  n — 1  fetters^ 
multiplied  by  ;?,  the  number  of  letters  employed.  Letting  Q  de- 
note the  number  of  permutations  of  n — 1  letters,  then  the 
general  formula  for  the  permutations  of  n  letters  will  be 
Qxn, 

24 


•  • 


278  ELEMENTS    OF   ALGEBRA.  [sECT.  X. 

If  71=: 2,  the  number  of  permutations  will  be  1x2=2. 

If  7^  =  3,  the  number  of  permutations  will  be  1x2x3=6. 

If  7^=:4<,  the  number  of  permutations  will  be  1x2x3x4' 
=24. 

If  7iz=5jthe  number  of  permutations  will  be  1x2x3x4 
X5=120,  6cc. 

Hence,  for  the  permutation  of  any  given  number  of  letters 
or  numbers,  we  infer  the  following  general 

RULE. 

Multiply  in  order  the  natural  numbers  1,  2,  3,  4,  <S*c.,  to  the 
number  denoting  the  letters  employed  inclusive  ;  the  result  will 
be  the  permutations  of  the  given  number  of  letters. 

EXAMPLES. 

1.  How  many  permutations  can  be  made  of  the  first  6  let- 
ters of  the  alphabet  1  Ans.  Ix2x3x4x5x 6 =720. 

2.  How  many  permutations  can  be  made  of  the  first  8  let- 
ters of  the  alphabet  ] 

3.  In  how  many  different  ways  may  12  different  persons 
be  seated  at  the  table  % 

II.     ARRANGEMENTS. 

413.  Arrangements  are  the  results  obtained  by  writing  a 
given  number  of  letters  in  sets,  2  and  2,  3  and  3,  &;c.,  in 
every  possible  order. 

Let  it  be  required  to  arrange  the  three  letters, 
a,  J,  and  c,  in  sets  of  two  each.  Setting  apart  ai 
we  write  after  it  each  one  of  the  reserved  letters, 
h  and  c,  and  thus  form  two  of  the  arrangements 
sought,  viz.,  ah  and  ac  ;  next,  setting  apart  Z*,  we  <{  be 
write  after  it  each  one  of  the  reserved  letters  a  and 
c,  and  form  two  more  of  the  arrangements  sought, 
viz.,  ba  and  be  /  pursuing  the  same  course  with  c, 
we  obtain      -------- 


ab 
ac 
— a 
ba 


The  arrangement  of  the  same  letters  in  sets  of 
one  each  would  give     ------        y 

Hence,  the  arrangement  of  three  letters,  taken  two  in  a  set,  will 


—b 
ca 
ch 
— c 
a 


SECT.  X.]        PERMUTATIONS    AND    COMBINATIONS.  279 

he  equal  to  the  arrangement  of  the  same  letters  taken  one  at  a 
time^  multiplied  by  the  number  of  letters  reserved. 

414.  Let  it  be  required,  in  the  next  place,  to  form  the  ar- 
rangement of  four  letters,  a,  i,  c,  and  </,  taken  three  in  a  set. 

First  arranging  the  letters  two  in  a  set,  we  shall  have  12 
arrangements,  viz. : 

oA,  ac^  ad,  ba,  be,  bd,  ca,  cb,  cd,  da,  db,  dc. 
Next,  take  one  of  the  above  sets,  ab,  for  example,  and  write 
after  it  successively  each  one  of  the  reserved  letters  c  and 
d,  and  thus  form  two  of  the  arrangements  sought,  viz.,  abc 
and  abd.  Proceeding  in  the  same  manner  with  the  remain- 
ing sets,  we  shall  obtain  24  arrangements,  viz. : 


abc 

bar. 

cab 

dab 

abd 

bad 

cad 

dac 

acb 

bca 

cba 

dba 

acd 

'      bed 

cbd 

dbc 

adb 

bda 

cda 

dca 

adc 

bdc 

cdb 

deb 

Hence,  the  arrangements  of  four  letters,  taken  three  in  a  set, 
will  be  equal  to  the  arrangements  of  the  same  letters  taken  two  in 
a  set,  multiplied  by  the  numler  of  letters  reserved. 

415.  In  like  manner,  we  have  the  arrangements  of  any  num- 
ber (m)  of  letters  taken  n  in  a  set,  equal  to  the  arrangements  of 
the  same  letteis,  n — 1  in  a  set,  multiplied  by  the  number  of  let' 
ters  reserved, 

416.  Let  P  represent  the  total  number  of  arrangements  of 
m  letters  taken  n—1  in  a  set,  supposing  this  number  to  be 
known  ;  the  reserved  letters,  when  it  is  required  to  take  n 
in  a  set,  will  be  m— (n  — l)=m— ti+I,  and  the  number  of 
arrangements  of  rn  letters,  taking  n  in  a  set,  will  be 

Px(w-7i+l). 
This  is  the  general  formula  for  arrangements.     To  apply 
it  to  particular  cases,  let  n=2;  then  m— n  + l=jn--l,  and 
P  will  represent  the  arrangements  of  m  letters  taken  1  at  a 


280  ELEMENTS    OF    ALGEBRA.  [sECT.  X. 

time;  whence  P=77j,  and  m{m  —  l)  will  represent  the  ar- 
rangements of  m  letters  taken  two  in  a  set. 

Again,  let  7^=3,  then  m—n-\-lz=zm—%  and  V—m{m — 1)  ; 
whence  the  formula  becomes 

m[m  —  \)  (77i  — 2). 
Let  71=^^  then  m—n-[-l=m—'6^  and  Vz=m(m—\)  (m—2)  ; 
whence  the  formula  becomes 

m(m  —  l)  (77^-2)  (m  —  S). 
Hence  we  infer  the  following  general 

RULE. 

1.  From  the  number  denoting  the  given  letters^  subtract  suc- 
cessively the  natural  numbers  1,  2,  3,  <^c.,  to  the  number  which 
denotes  the  letters  to  be  taken  at  a  time. 

2.  Multiply  these  several  remainders  and  the  number  denoting 
the  given  letters  together ;  the  product  will  be  the  arfangements 
required. 

EXAMPLES* 

1.  How  many  arrangements  may  be  made  of  the  first  six 
letters  of  the  alphabet,  taken  three  in  a  set  % 

In  this  example  m  —  Q,  and  n  =  3;  then  we  have 
77^(m— 1)  (77^  — 2)  =  6x5x4==120.     ^ns. 

2.  How  many  arrangements  may  be  made  of  the  26  letters 
of  the  alphabet,  taking  6  in  a  set  1 

Jsote. — It  should  b«  observed  that,  when  m^=Ln^  the  num- 
ber of  arrangements  is  the  same  as  the  number  of  permuta- 
tions. Thus,  if  there  be  six  letters,  to  be  taken  six  in  a  set, 
we  have 

77i(7?i-l)(m-2)(m-3)(m— 4)  (?7i-5)  =  6x 5x4x3x2x1== 
720. 

III.    COMBINATIONS. 

417.  Combinations  are  arrangements,  any  two  of  which 
will  differ  from  each  other  by  at  least  one  of  the  letters 
which  enter  into  them. 

Let  it  be  required  to  determine  the  number  of  combina- 
tions of  which  the  three  letters  a,  5,  and  c^  taken,  two  in  a 


SECT.  X.]         PERMUTATIONS    AND    COMBINATIONS.  281 

set,  are  susceptible.     The  arrangements  of  these  letters,  two 
in  a  set,  are 


ab  ac  be 

ba  ca  cb 

In  these  six  arrangements  we  have  but  three  combina- 
tions, viz.,  aby  aCy  and  ic,  each  one  of  which  is  repeated  as 
many  times  as  there  are  permutations  of  two  letters. 

Hence  the  combinations  of  three  letters^  taken  two  in  a  set, 
will  be  equal  to  the  arrangements  of  three  letters,  taken  two  in  a 
sety  divided  by  the  permutations  of  two  letters. 

4-18.  In  like  manner,  it  may  be  shown  that  the  combina- 
tions of  four  letters,,  taken  three  in  a  set,  are  equal  to  the 
arrangements  of  four  letters,  taken  three  in  a  set,  divided 
by  the  permutations  of  three  letters. 

And,  in  general,  the  combinations  of  m  letters,  taken  n  ia 
a  set,  will  be  equal  to  the  arrangements  of  m  letters,  taken 
71  at  a  time,  divided  by  the  permutations  o(  n  letters. 

Hence  we  have  the  following  general  formula  for  combii* 
nations: 

Px(m~-7t+l) 
Qxn 

Letting  »=2,  the  formula  for  the  combinations  of  m  let- 
ters, 2  in  a  set,  becomes 

m(m — 1) 
1x2 
Letting  7i=3,  the  formula  for  the  combinations  of  m  let^ 
ters,  taken  3  in  a  set,  is 

m{m^\)  (m— 2) 
1x2x3 
And,  if  71=4,  we  shall  have 

m{m—\)  (7»— 2)  (7»— 3) 
1x2x3x4 
Hence  we  infer  the  following  general 

RULE.. 

To  find  the  combinations  of  m  letters,  taken  n  in  a  set, 

Nn 


289  ELExMBNTS    OF    ALGEBRA.  [sECT.  X, 

divide  the  arrangements  of  m  letters^  taken  n  in  a  set,  by  the 
permvtations  of  n  letters, 

EXAMPLES. 

1.  How  many  combinations  can  be  made  of  10  letters,  ta- 
king foar  in  a  set  % 

;7^(m-l)  (771-2)  (77^—3)_  10x9  x8x7_^^^      ^^^^ 
QX7?J  1x2x3x4 

2.  How  many  combinations  can  be  made  of  the  26  letters 
of  the  alphabet,  taking  two  in  a  set  1 

3.  How  many  combinations  can  be  made  of  100  things, 
tsfking  four  in  a  set  1  ' 

GENERAL  DEMONSTRATION  OF  THE  BINOMIAL 
THEOREM. 

419.  We  hare  already  exhibited  the  Binomial  Theorem 
and  its  applications  to  numbers  ;  but  we  propose  now  to  give 
a  more  rigid,  and-,  at  the  same  time,  a  more  general  demon- 
stration of  it.  It  is  easy  to  fix  upon  the  law  for  the  expo- 
nents, but  that  for  the  coefficients  is  not  so  obvious.  If  we 
observe,  however,  the  manner  in  which  the  different  terms 
that  compose  a  power  are  formed,  we  shall  perceive  that  the 
numerical  coefficients  are  occasioned  by  the  reduction  of 
several  similar  terms  into  one,  and  that  these  similar  terms 
arise  from  the  equality  of  the  factors,  which  compose  a 
power.  Hence,  these  reductions  will  not  take  place  if  the 
second  terms  of  the  binomial  are  diffigrent. 

420.  We  will  begin,  therefore,  by  investigating  the  law 
for  the  formation  of  the  prodtict  of  any  number  of  binomi- 
als, a?4-a,  a?-f-i,  a?+c, of  which  the  first  term  is  the 

same  in  each,  and  the  second  terms  diffi?rent.  The  coeffi- 
cients of  like  powers  of  x  are  placed  under  each  other,  and 
separated  from  a:,  into  which  their  sum  is  to  be  multiplied, 
by  a  vertical  line. 


X  -\-a 
X  -tb 


1st  product  .  .     x^-\-a 


X  -i-ab 


SECT.  X.] 


BINOMIAL    THEOREM. 


28S 


X  -f  c 


2d  product 


3d  product 


x^-^a 

x'-ifah 

+  b 

+  ac 

+  c 

^hc 

X  -^abc 


x+d 


x*-\-a 

x'+ab 

a^-^abc 

+  b 

4-ac 

+  abd 

+  c 

-^ad 

-{-acd 

+t/ 

+  cd 

+  bcd 

x-i-abcd 


4f'21.  From  an  incpectron  of  the  above  products,  which  we 
have  formed  by  the  common  rules  of  multiplication,  it  will 
be  easy  to  infer  the  following  laws : 

1.  The  exponent  of  x  in  the  first  term  is  the  same  as  the  num- 
ber of  binomial  factors  employed^  and  decreases  by  1  in  each  of 
the  following  terms. 

2.  The  coefficient  of  the  first  term,  is  unity  ;  the  coefficient  of 
the  second  term  is  equal  to  the  s-um  of  the  second  terms  of  the 
binomials  ;  that  of  the  third  term  is  equal  to  the  sum  of  the  dif- 
ferent combinations  of  the  second  terms  of  the  binomials^  taken 
two  in  a  set  y  and  that  of  the  fourth  is  equal  to  the  sum  of  the 
products  of  the  second  terms  of  the  binomials^  taken  three  in  a 
56/,  and  so  on.  The  last  term  is  equal  to  the  product  of  the  sec- 
ond terms  of  the  binomials. 

422.  In  order  to  show  that  the  same  law  will  obtain  what- 
ever be  the  number  of  factors  employed,  it  is  only  necessary 
to  prove  that  if  the  law  be  true  for  the  product  of  any  num- 
ber (m)  of  binomials,  it  will  also  be  true  for  the  product  of 
m-\-\  binomials. 

Let  us  represent  the  product  of  m  binomial  factors  by 
ar'^-hAx''-'4-Bx'«-'+Cx"-'+ U.      m 

Multiplying  this  expression  by  a  new  factor,  x+K,  it  be- 
comes 


+  K 


X-+B 

+AK 


I— '4-0 
4BK 


tKL 


284 


ELEMENTS    OF    ALGEBRA. 


[sect.  X. 


423.  Here  the  law  of  the  exponents  is  evidently  the  same 
as  before.     With  respect  to  the  coefficients,  it  is  evident, 

1.  That  the  coefficient  of  the  first  term  is  unity. 

2.  That  A-j-K,  the  coefficient  of  the  second  term,  is 
equal  to  the  sum  of  the  second  terms  of  the  m-\-l  bino- 
mials. 

3.  That,  since  B,  by  hypothesis,  expresses  the  sum  of  the 
second  terms  of  the  m  binomials,  taken  two  in  a  set,  and 
AK  expresses  the  sum  of  the  second  terms  of  the  m  bino- 
mials, multiplied  each  by  the  new  second  term  K,  B-j-AK, 
the  coefficient  of  the  third  term  will  be  the  sum  of  the  prod- 
ucts, two  in  a  set,  of  the  second  terms  of  the  m-\-l  bino- 
mials. 

4.  That  C-f  BK  is  the  sum  of  the  products,  taken  three 
in  a  set,  of  the  second  terms  of  the  m-\-l  binomials,  and  so 
on. 

5.  That  the  last  term  UK  is  the  product  o(  m  +  1  second 
terms. 

424.  The  law  laid  down  in  Art.  421*,  being  true  for  ex- 
pressions of  the  fourth  degree,  will,  from  what  has  just  been 
demonstrated,  he  true  for  those  of  the  fifth  ;  and,  being  true 
for  expressions  of  the  fifth  degree,  will  also  be  true  for  those 
of  the  sixth,  and  so  on  indefinitely. 

425.  If,  in  the  different  products  which  we  have  formed^ 
we  make  the  second  terms  of  all  the  binomials  equal,  i.  e.y 
make  a=J=c  =  ti,&c.,  these  products^  will  be  converted  into, 
powers  of  a;+a.     Thus, 


X  -{-a 
X  -\-a 


1st  product 


2d  product 


x^-^a 
X  -\-a 


X  -\-o^ 


3?^-a 

a?2+a^ 

+« 

+  a^ 

+fl^ 

+«^ 

X  -\-(i 


SECT.  X.] 


BINOMIAL    THEOREM. 


285 


3d  product 


x-\-a 

x*+a 

or' -{-a' 

a:Ha' 

+  a 

+a» 

+  «> 

+  a 

+a» 

H-a* 

-f» 

+  0' 

x+o* 


426.  By  comparing  these  results  with  the  products  from 
which  they  have  been  derived,  we  perceive, 

1.  That  the  multiplier  of  x  in  the  second  term  has  been 
converted  into  the  first  power  of  a,  repeated  as  many  times 
as  there  are  units  in  the  number  of  binomial  factors  used, 
or,  which  is  the  same  thing,  as  there  are  units  in  the  expo- 
nent denoting  the  power  to  which  x-\-a  was  to  be  involved, 

2.  That  the  multiplier  of  x  in  the  third  term  has  been 
converted  into  a^  repeated  as  many  times  as  there  can  be 
formed  different  products  from  a  number  of  letters  equal  to 
the  number  of  binomials  employed,  taken  two  in  a  set. 

3<  That  the  multiplier  of  the  fourth  term  has  been  con- 
verted into  a',  repeated  as  many  times  as  there  can  be  form- 
ed different  products  from  a  number  of  letters,  equal  to  the 
number  of  binomials  employed,  taken  three  in  a  set,  and  so 
on. 

427.  It  is  therefore  evident  that,  whatever  may  be  the 
power  to  which  the  binomial  a:  +  a  is  to  be  raised,  the  for- 
mation of  its  power  will  be  subject  to  the  following  laws, 
viz. : 

1.  The  exponent  of  x  in  the  first  term  will  be  equal  to  the  ex- 
ponent  of  the  power ,  and  in  the  succeeding  terms  will  decrease 
regularly  by  1  to  the  last  teivi^  in  which  it  will  le  0, 

2.  The  exponent  of  a  in  the  first  term  will  be  0,  in  the  second 
1,  and  that  it  will  go  on  increasing  by  1  until  it  becomes  equal 
to  the  exponent  of  the  power  to  which  the  binomial  was  to  be  in* 
volved. 

3.  That  the  numerical  coefficient  of  x  in  the  first  term  will  b€ 
1 ;  in  the  second  it  will  be  equal  to  the  exponent  denoting  th% 


S8^         >  ELEMENTS    OF   ALGEBRA.  [sECT.  X. 

power  to  which  the  binomial  was  to  le  involved ;  in  the  third 
term  it  will  be  equal  to  the  number  of  products^  which  may  be 
formed  from  a  number  of  letters^  equal  to  the  exponent  denoting 
the  power  of  the  binomial,  taken  two  in  a  set ;  in  the  fourth  term 
it  will  be  equal  to  the  numbev  of  products  which  may  be  formed 
from  the  same  number  of  letters  taken  three  in  a  set,  Sfc.     ♦ 

428.  The  above  theorem,  with  reference  to  the  coeffi- 
cients, is  too  cohiplicated  for  general  use.     In  order  to  sini- 
plify  it,  let  it  be  required  to  expand  (oj+a)"*.     The  first  few. 
and  the  last  few  terms,  without  the  numerical  coefficients, 
will  be 

+a^     (A). 
The  numerical  coefficient  of  the  first  term  is  1 ;  that  of 
the  second  is  m  /that  of  the  third  is  equal  to  the  number  of 
products  which  may  be  formed  of  m  letters  taken  two  in  a 

set ;  this  is  expressed  by  the  formula  —A 1 :  the  coeffi- 
cient of  the  fourth  term  is  — —       ' ' — — ^ ,  &;c. 

1x2x3 

By  inspecting  the  above  formulas  for  the  numerical  coef- 
ficients of  X,  it  will  be  perceived  that  the  coefficient  of  the 
third  term  is  equal  to  the  coefficient  of  the  second  term  (m) 
multiplied  by  the  exponent  of  x  {m — 1)  in  that  term,  the 
product  divided  by  the  number  (2)  which  marks  the  place 
of  this  term,  counting  from  the  left. 

And,  also,  the  coefficient  of  the  fourth  tferm  is  equal  to 

the  coefficient  of  the  third  term  /  ^(^—  ;  \  ^  multiplied  by 

the  exponent  of  x  (m — 2)  in  that  term,  the  product  divided 
by  the  number  (3)  denoting  the  place  of  that  term  &c. 

429.  Again,  since  in  the  expression  [x-{-a)  ,  a  may  be  sub- 
stituted for  X,  and  x  for  a,  without  altering  its  value,^  it  fol- 
lows that  the  same  thing  may  be  done  in  the  development 
of  it.  Hence,  if  this  development  contains  a  term  of  the 
form  Ka"a?"'~"  (K  representing  the  numerical  coefficient),  it 


SECT.  X.]  BINOMIAL   THEOREM.  287 

must  have  another  equal  to  Kx^aT-''^  or  Ka'^'^x".  These 
two  terms  are  evidently  at  equal  distances  from  the  two 
extremes,  for  the  number  of  terms  whioh  precede  any  term 
being  indicated  by  the  exponent  of  a  in  that  term,  it  follows 
that  the  term  Ka"x"*~"  has  n  terms  before  it,  and  that  the 
term  Ka^^x"  has  m — n  terms  before  it,  and,  consequently,  n 
terms  after  it,  since  the  whole  number  of  terms  is  denoted 
by  m-fl. 

Therefore,  in  the  development  of  any  power  of  a  binomial, 
the  coefficients  at  equal  distances  from  the  extremes  are  equal  to 
each  other. 

Hence,  the  numerical  coefficients  of  the  series  A  will  be 

l-LffT-L^^— 0-j-K^— 1)  (^— -)  ■  m{m—l)  (ct— 2) 

1x2  1x2x3        ^  '  '  *' '         1x2x3 

+!fcL)+m+l     (B). 

Compounding  the  two  series  A  and  B,  we  have 

(x+ar=x--\-max-'  +  ^lil^^^a'x"-'  +  m(m-l)  (m-2)^ 
^  ^  ^  1x2  1x2x3 

x-'-f   .  .  .   4.^(^-1)  (^-2)^^.^^^(m-l)^^      , 
1x2x3  1x2 

4-mxa"'~'  +  a"'. 

430.  The  preceding  operations  give  rise  to  the  following 
simple  theorem  for  obtaining  the  coefficients : 

1.  The  coefficient  of  the  first  term  is  1  j  that  of  the  second  is 
equal  to  the  number  of  units  in  the  exponent,  which  denotes  the 
power  to  which  the  binomial  is  to  be  raised. 

2.  .^nd  univp-sally^  if  we  multiply  the  numerical  coefficient 
by  the  exponent  of  x  in  that  term,  and  then  divide  the  product  ly 
the  number  which  marks  the  place  of  that  term  from  the  left,  the 
quotient  will  be  the  coefficient  of  the  succeeding  term. 

3.  The  terms  in  the  last  half  of  the  series  of  coefficients  will 
be  found  to  correspond  with  those  in  the  first  half  placed  in  the 
inverse  order. 

431.  These  results  of  the  Binomial  Formula  are  substanr 
tially  the  same  as  those  obtained  by  a  different  process,  and 
practically  applied  in  Articles  259-272. 


288  ELEMENTS    OF   ALGEBRA.  [SECT.  X. 

It  should  also  be  remarked  that  the  same  formula  will 
apply  whether  m  represent  a  positive  or  negative  whole 
number  or  a  fraction. 

CONTINUED  FRACTIONS. 

432.  A  continued  fraction  is  one  which  has  1  for  its  nu- 
merator, and  for  its  denominator  an  entire  number  plus  a 
fraction  \  which  fraction  also  has  1  for  its  numerator,  and 
for  its  denominator  an  entire  number  plus  a  fraction,  and  so 
on.     Thus, 

1 

«+l 

c+l 


</+,  &c., 
is  a  continued  fraction. 

I.  To  convert  a  vulgar  fraction  into  a  continued  fraction. 

RULE. 

Jjpply  to  the  two  terms  of  the  fraction  the  process  of  finding 
their  greatest  common  divisor  ;  continue  the  operation  until  0  is 
obtained  for  a  remainder  ;  the  reciprocals  of  the  successive  quo- 
tient will  form  the  partial  fractions,  which  constitute  the  contin^ 
ued  fraction. 

JSTote. — The  above  rule  may  be  readily  illustrated  by  ap- 
plying it  to  a  particular  case.     Take,  for  example,  the  frac- 

351 
tion ■ ;  dividing  both  numerator  and  denominator  by  the 

965  '  ^  ^ 

numerator,  we  obtain 

351     1 


965     2+263 
351* 

263 

Performing  the  same  operation  upon ,  we  obtain 

dO  1 

263     1 


351     1+88 
263' 


SECT.  X.]  CONTINUED    FRACTIONS. 

A     •  88      1 

Again 


And 


263 

2+87 

88' 

87 

1 

88 

1+  1. 

87' 

351 

1 

965 

2+1 

1  +  1 

2+1 

Hence 


1+J^ 

87* 
Now,  if  we  apply  the  or4^nary  rule  for  finding  the  great- 
est common  measure  of  two  numbers  to  the  two  terms  of 

351 
the  fraction ,  the  successive  quotients  will  be  2,  1,  2,  1, 

87,  and  their  reciprocals  ^,  |,  ^,  i,  and  ^V^,  which  are  evi- 
dently the  partial  fractions  which  compose  the  above  con- 
tinued fraction. 

EXAMPLES. 

65 
1.  Transform  —  into  a  continued  fraction. 
149 


149     2+ 1 


3+1 

2+1 


2+1 


1+1 

2' 

2.  Transform into  a  continued  fraction. 

5537 

"  965 

3.  Transform  —L  into  a  continued  fraction. 

3ol 

4.  Transform  — - —  into  a  continued  fraction. 

10948 

11.  To  find  the  equivalent  vulgar  fraction  for  a  given  con- 
tinued fraction. 

25  Oo 


• 


290  ELEMENTS    OF    ALGEBRA.  [sECT.  X. 

•  RULE. 

1.  If  there  he  any  whole  number  prefixed  to  the  fractional 
series^  that  will  be  the  first  approximate  value  ;  if  there  be  no 
such   whote  number,  then  we  know  that  the  vulgar  fraction 

sought  is  proper,  and  the  symbol  _  is  used  to  express  its  first 

approximate  value. 

2.  The  second  approximate  value  is  obtained  by  taking  the 
sum  of  the  first  approximate  value  and  the  first  partial  fraction, 

3.  To  obtain  the  third  approximate  value,  multiply  the  numer- 
ator and  denominator  of  the  second  approximate  value  by  the  de- 
nominator of  the  next  partial  fraction,  and  to  the  respective 
products  add  the  numerator  and^  denominator  of  the  first  ap- 
proximate value. 

4.  Jlnd  universally,  if  we  multiply  the  terms  of  the  last  ap- 
proximate value  by  the  denominator  of  the  succeeding  partial 
fraction,  and  to  the  products  add  the  numerator  and  denominator 
of  the  preceding  approximate  value,  the  result  will  be  the  suc- 
ceeding approximate  vaiue.  Thus  continue  till  the  last  partial 
fraction  has  been  used. 

Jfote  1. — The  preceding  rule  may  be  readily  illustrated  by 
applying  it  to  a  particular  example.  Thus,  let  it  be  requi- 
red to  find  the  equivalent  vulgar  fraction  for  the  continued 
fraction 


1  2+i 


3+1 


2+1 


2+1 


1+i 

2' 


Here  the  first  approximate  value  is       -        - 
The  second,  omitting  all  after  the  first  partial 

1  ■  2""2 


fraction,  is _+_— . 


The  third,  omitting  all  after  the  first  tWo  par- 


tial fractions,  is 


SECT.  X.]  CONTINUED    FRACTIONS.  291 


0  1       _0     1 

1  2+\     1^2x3+1 

,orl><3+0    : 
'2x3+1 

_  3 

7' 

3 

The  fourth  is  ?+i_ 

3x2+1 
7x2+2 

7 
16" 

3+1 

2 

The  fifth  is     2+L^ 

7x2+3 

16x2+7 

17 
39' 

3+1 

2+1. 

2 

The  sixth  is   5+L_ 

_  17xl+7_ 
39x1+16 

24 
■  55 

3+1 

2+1 

2+k 

The  seventh  is  ^+^ 

i 

_24x2+17_ 
55x2+39 

.65 
149 

3+1 

.2+1 

- 

2+1 

1+1 

2 

^ote  2. — The  successive  reductions,  it  will  be  perceived 
by  inspecting  the  above  results,  are  alternately  less  and 
greater  than  the  whole  continued  fraction,  and  they  ap- 
proximate this  fraction  nearer  find  nearer.  The  first  re- 
duction is  always  less  than  the  whole  continued  fraction. 
Hence  the  reductions  of  an  odd  rank  are  always  less  than  the 
whole  continued  fraction^  and  those  of  an  even  rank  are  greater. 

In  the  above  reductions. 

The  second  difi^ers  from  the  true  value  of  the  continued 

1      65  19 


fraction  by     - 
The  third  differs  by 


2     149  ~  298 

66  __  3  __    8 

149      7  1043' 


.LGEI 

5RA. 

[sect.  X, 

7 

le- 

65 

149 

3 

2384 

es 

17 

2 

149 

39 

5811 

24. 
55 

e5 

149 

_  1 

8195 

e5 

149 

es 

149 

=   0. 

292  ELEMENT 

The  fourth  diifers  by    - 
The  fifth  differs  by 
The  sixth  differs  by 

The  seventh  differs  by  - 

J^Tote  3. — If  a  vulgar  fraction  which  is  not  expressed  in  its 
lowest  terms  be  converted  into  a  continued  fraction,  and  all 
the  reductions  be  formed  to  the  last  inclusive,  the  last  re- 
daction will  not  be  the  proposed  fraction,  but  this  fraction 
reduced  to  its  lowest  terms. 

348 

For  example,  let  the  fraction  be  converted  into  a 

^  954 

continued  fraction.     Thug, 

348^0     1 


954     12+1 


1+1 


1+1 


1+1 

The  reductions  of  this  continued  fraction  are, 
0    1    1    2    3         ,  29 
1'  2'  3'  5'  8'  W 

29   .  348 

The  last  reduction,  — ,  is  the  same  as reduced  to  its 

77  954 

lowest  terms. 

EXAMPLES. 

1.  Required  the  vulgar  fraction  which  is  equivalent  to  the 
continued  fraction 

1 

3+T 


2+1 


5+1 


6  992 


SECT.  X.]  CONTINUED   FRACTIONS.  2^3 

2.    Required  the   approximate  values  of  the   continued 
fraction 

1 

IH-l 

2+1 


3+1 


4+1 


5+1 

6+1 


7+1 


8+1 

3.  Required  the  approximative  values  of  the  continued 
fraction 

1 

9+1 

8+1 


7+1 


6+1 


5+1 


4+1 


3+1 
2- 

4.  The  ratio  of  the  circumference  of  a  circle  to  its  diam- 

eter  may  he  expressed  hy  the  fraction  ^^7^;^.^-^  >   required 

some  of  the  approximative  values  of  this  ratio. 

Converting  the  given  fraction  into  a  continued  fraction^ 
we  have 

314159^3^1 


100000  7+1 

15+1 

1  +  1 

25+j 

1+1 

7+1 
4* 


994l  elements  of  algebra.  [sect.  x. 

The  successive  reductions  are, 

3    22    333    355    9208    9563    76149    ^^^  314159' 
r    7'  106'   113'  2931'  3044'  24239'  ^"     lOOOOO' 

INFINITE  SERIES. 

433.  An  Infinite  Series  is  a  progression  of  numbers  con- 
nected together  by  the  signs  +  or  — ,  proceeding  onward 
without  termination,  but  usually  according  to  some  regular 
law,  which  may  be  discovered  by  tracing  a  few  of  the  lead- 
ing terms. 

A  Converging  Series  is  one  whose  successive  terms  de- 
crease.    Thus, 

-+-2  +  3+4  +-5  +r^^'> 

X     x^     a^     x^      x^ 

and        -        -       1+A +1+2+^4-, &c., 
2     4      8      16     32     '       ' 

are  converging  series,  when  a?>l  in  the  first  series. 

A  Diverging  Series  is  one  whose  successive  terms  increase. 
Thus, 

x-\-x^-{-x'-{-x*  +  x'-\-,  &c., 

and        -  2-1-4,  4-8-1- 16+32+,  &c., 

are  diverging  series,  when  a7<  1  in  the  first  series. 

I.    EXPANSION  OF  INFINITE  SERIES.. 

434.  There  are  four  general  methods  of  converting  alge- 
braic expressions  into  an  infinite  series  of  equivalent  value. 

First.  We  have  already  seen  that  the  division  of  alge- 
braic quantities  (Art.  110)  will  sometimes  produce  an  infi- 
nite series.  Also,  a  fraction  may  sometimes  be  expanded 
into  an  infinite  series  by  dividing  the  numerator  by  the  de- 
nominator. 

examples.. 
1.  Divide  1+a  by  1 — a. 


SECT.  X.]  EXPANSION    OF    INFINITE    SERIES.  899 

1— a  l4-2a4-2a'+2a»+2a*-f ,  &c.,  ad  infin. 


2a— 2a« 


2a'— 2a« 


2a» 
2a*— 2a* 

2^ 

2.  Reduce  the  fraction to  an  infinite  series. 

1 — a 

Since  the  value  of  a  fraction  is  the  quotient  resulting  from 

the  division  of  the  numerator  by  the  denominator  (Art.  125), 

the  vahie  of  the  above  fraction  will  be  obtained  by  dividing 

1  by  1— a. 

1       |1— fl 

1 — a  l-|-o+a'4-a*+fl^-f  >  &c.,  ad  inftn, 
a 


a^—d? 


a' 


KoH, — By  observing  that  the  value  of  a  fraction  is  equal 
to  the  terms  of  the  quotient  -|-  the  fraction  formed  by  pla- 
cing the  remainder  over  the  denominator^  we  shall  have 

--i_  =  l+a+a»+a»-fa'+a*+ a"+:^. 

1— a  \—a 

3.  Reduce  the  fraction to  an  in£aite  series. 


296  ELEMENTS    OF    ALGEBRA.  [sECT.  X. 

4.  Reduce  the  fraction to  an  infinite  series. 

a—-b 

a      or      a^ 

5.  Reduce  the  fraction to  an  infinite  series. 

1  +  a 

Ans.  \—a-\-d!' — o^-\-a^ — a^,  &c.,  adinjin, 

6.  Reduce  the  fraction^ to  an  infinite  series. 

1-2 

Ans.  1  +  2+4+8+ 16  +  32+64+,  &c.,  adinfin. 
J^ote. — The  above  result  might,  at  first  sight,  seem  ab- 
surd ;  but  it  should  be  remarked  that,  if  we  wish  to  stop  at 
any  term  of  the  above  series,  we  must  add  the  fraction  that 
remains  to  the  terms  taken.  Thus,  if  we  stop  after  taking 
seven  terms  of  the  quotient,  we  shall  have 

1        1  +  2+4+8+ 16  +  32+64+-!??-=:  127+1^=— 1. 


1_2~  \        1-2  —1 

7.  Reduce  the  fraction  -^ —  to  an  infinite  series. 

a—x 

8.  Reduce  the  fraction .  to  an  infinite  series. 

435.  Secondly.  An  infinite  series  may  be  formed  by  ex- 
tracting the  root  of  a  compound  surd. 

EXAMPLES. 


1.  Reduce  -s/ c^-\-h^  to  an  infinite  series. 
Extracting  the  square  root,  according  to  the  rule  given  in 
Art.  291, 


a2+^,^(a+^__|l+JL,  &c.,  adinfin, 
2a     8a'     16a' 


2aY 


SECT.  X.]  EXPANSION    OF    INFINITE    SERIES.  297 


2.  Reduce  V<^—t^  to  an  infinite  series. 


3.  Reduce  ^/l^x  to  an  infinite  series. 

Thirdly.  We  have  already  seen  (Art.  295)  that  if  a  bino- 
mial which  has  a  negative  or  fractional  index  be  expanded  by 
the  Binomial  Theorem,  it  will  produce  an  infinite  series. 

This  case  has  already  been  sufficiently  explained  and  il- 
lustrated in  the  article    referred  to  above. 

436.  Foitrthly,  An  algebraic  expression  may  also  be  ex- 
panded by  assuming  a  series  with  indeterminate  coefficients^- 
and  afterward  finding  the  value  of  these  coefficients. 

To  give  some  idea  of  this  method  of  development,  we  will 

suppose  it  is  required  to  expand  — ^  into  a  series  arran- 
ged according  to  the  ascending  powers  of  x.  This  expres- 
sion may  evidently  be  expanded  so  as  to  answer  these  con- 
ditions ;  for _=a(c-f  ia?)"*.     Expanding  this  last  expres- 

sion  by  the  binomial  theorem,  and  representing  the  known 
terms  and  coefficients  successively  by  A,  B,  C,  &c.,  we  shall 
have 

=A4-Bx-fCa:2-fDx*+Ex*-f ,  &c.,  adinjin. 


c-\-bx 

The  above  coefficients  A,  B,  C,  &c.,  being  functions  of 
a,  by  and  c,  that  is,  dependant  on  them  for  their  values,  but 
independent  of  x,  are  called  indeterminate  coefficients. 

It  is  now  required  to  determine  thp  value  of  these  coeffi- 
cients. 

Multiplying  both  members  of  the  equation  by  the  denom- 
inator c-j-^^j  and  transposing  a,  we  obtain 

0={Ac-a)-\-Ab     x-{-Bb  I  x^+Cb     r'+j  &c.,  adinjin, 

+Bc        +Cc  I      -f-Dc 
Here  it  is  evident  that  if  Ac — a,  A6+Bc,  Bi+Cc,  &c.,  be 
made  each  equal  to  0,  the  several  terms  of  the  second  mem- 
ber will  be  reduced  to  0,  and,  consequently,  the  mendber  will 

Pp 


298  ELEMENTS    OF    ALGEBRA.  [SECT.  X. 

equal  0,  and  thus  the  conditions  of  the  equation  may  be  sat- 
isfied. From  the  above  assumption  we  derive  the  following 
values  of  the  successive  coefficients  : 

1st     -      Ac— a  =0;  hence  A  —  -. 

c  ♦ 

2d      -      Ah+Bc=0',henceB=-^=-^X     ":  =-^. 

c  c         c  c^ 

3d      -      BJ+Cc=:Oj  henceC=-i^=-^X-^=:.  +  -. 

c  c         c^  a? 

4th     -       CZ.+Dc=0ihenceD  =  -2^=-^X^'    =-^'. 


Hence  we  have 


— ^-f-__ar— __a; ,  &c.,  ad  mjin. 


c-\-bx     c     c         c^  c'* 

437.  By  inspecting  the  preceding  operations,  we  shall  per- 
ceive that  each  succeeding  coefficient  is  equal  to  the  prece- 
ding multiplied  by  — -  ;  consequently,  — -  is  the  ratio  of  the 
c  c 

progression  of  the  coefficients,  and  — —  is  the  ratio  of  the 

c 
progression  of  the  series. 

EXAMPLES, 

1.  Expand into  an  infinite  series. 

b—ax 

Assume  =A-\-Bx-\-Cc(^-{-'Dx'^-{-Ex'^-{-,  &c.,  ad  infin. 

b — ax 

Multiplying  both  members  of  the  equation  by  b — ax,  and 

transposing  c?,  we  have 


0=z(Ab—d)—Aa 

-{-Bb 


X — Ba 

+  Cb 


x^—Ca 
+Db 


oc^—Da 

+Eb 


x\  &c. 


Whence,  making  the  several  coefficients  equal  to  0,  we 
have, 

1st        '  Ab — d  =0;  hence  A=_. 

b 

id        .'         Bb—Aa=.0',  hence  B=^=  ^  X^=~. 

b        0      b      b' 

3d         -  C^-Ba:=0  ',  hence  C=^=^  X?^^. 

b      b^      b      b 


SECT.  X.]  EXPANSION   OF   INFINITE    SERIES.  299 

4th        -         DJ-Ca=Oi  hence  D=^=^x?=^. 

5th       -        Ei-Da=0;  hence  E=°?=t''x?=^. 

0        tr      0      br 

Hence  we  have 

2.  Expand  — "^         into  an  infinite  series. 

Ans.  l+3x-h4x'+7x*+llat*+18a:',  &c. 

3.  Expand  . =  into  an  infinite  series. 

Am.  l+a?+2a?'+2x»-H3a!:*-|-3x»+4a?«4-4a?',  &c. 
1— X 
1— 2x— 3x- 
Ans,  l+x4-5x'+13ar'4-41x*+121i»4-365x«,  &c. 


4«.  Expand  - — ^— ,  into  an  infinite  series. 


5.  Expand    ^"'"^  into  an  infinite 


series. 


Remark. — The  method  of  indeterminate  coefficients  re- 
quires that  we  should  know  the  form  of  the  development 
with  reference  to  the  exponents  of  x.  The  terms  are  gen- 
erally supposed  to  be  arranged  according  to  the  ascending 
powers  of  x,  commencing  with  x°.  Sometimes,  however, 
this  form  is  not  exact  \  in  this  case  the  calculus  detects  the 
error  in  the  supposition. 

For  example,  let  it  be  required  to  expand  the  fraction 

1 
Si-x"' 

Suppose  — L_-A+Ba:-hCx»+Da:*,  &c. 
3x — :r 

Multiplying  both  members  by  Zx — a?*,  and  transposing  1, 

we  have 


0=— l  +  3Ax— A 


Ix*, 
+  3B        -I-3C        +3D  [ 


x«-C    [x*,&c. 
+  3D 


x«— B 
-I-3C 

Whence  the  conditions  of  the  equation  require  that  — 1  =  0, 
which  is  absurd  j  hence  the  above  form  will  not  apply  to  the 

development  of  the  expression — . 

OX— x^ 


300  ELEMENTS    OF   ALGEBRA.  [SECT.  X. 

II.    SUMMATION  OF  INFINITE  SERIES. 

438.  The  summation  of  a  series  is  the  finding  a  finite  ex- 
pression equivalent  to  the  series. 

But  as  different  series  are  often  governed  by  very  differ- 
ent lavi^s,  the  methods  of  finding  the  sum  which  are  appli- 
cable to  one  class  of  series,  will  not  apply  universally.  Hence 
result  different  methods  of  summation. 

I.  First  Method. — If  the  series  is  a  regular  descending 
geometrical  series,  that  is,  if  its  terms  decrease  by  a  com- 
mon divisor,  the  sum  of  the  series  may  be  obtained  by  the 
following  formula  :     (Art.  389.) 

1—2' 
As  this  formula  has  been  explained  and  applied  (see  Arti- 
cles 380  to  39p)  in  Geometrical  Progression,  we  need  add 
nothing  more  concerning  it  in  this  place. 

II.  Second  Method. — The  summation  of  certain  classes 
of  infinite  series  may  be  effected  by  subtraction. 

EXAMPLES. 

1.  Let  it  be  required  to  find  the  sum  of  the  infinite  series 

'   ;+.4^+:r^.  +  rL+^.'&c.  (1.) 


1.2.3     2.3.4     3.4.5     4.5.6     5.6.7 
By  removing  the  last  two  factors  from  each  of  the  de- 
nominators in  the  preceding  series,  let  us  form  a  new  series 
whose  value  may  be  expressed  by  S  j  thus, 

S=i-fl-f  l+-+i+i  &c.,  ad  injin,  (2.) 

1-23456  ''  ^    ' 

By  transposition, 

^-\=l+\+\+l+\^''-^''^'''fi'>-  (3-) 

By  subtracting  the  last  equation  (3)  from  the  second  (2), 

1= — + — +  —  -j- — + — .  &cc.t  ad  infin.  (4.) 

1.2     2.3     3.4     4.5     5.6'        '         *^  ^    ^ 

By  transposition, 

1 — _= — + — +  — + — ,  &c.,  ad  iniin,  (5.) 

2    2.3     3.4    4.5     5.6'        '         '^  ^    ^ 


SECT.  X.]  SUMMATION    OP    INFINITE    SERIES.  801 

Whence,  by  subtracting  this  last  equation  (5)  from  the 
fourth  (4),  we  have 

!=_!_  +_^4— ^4— i5_,  &c.,  ad  infin, 
2     1.4.3    2.9.4     3.16.5     4.25.6' 

Or,  i=_?_-f_^-f -^  +-^,  &c.,  ad  infin. 
'2     1.2.3^2.3.4     3.4.5      4.5.6'  -^ 

Whence,  dividing  by  2, 

l=_L_+_J_-f-— 1-  +  -?— ,  &c.,  ad  infin. 
4     1.2.3     2.3.4     3.4.5     4.5.6'  -^ 

Hence  the  sum  of  the  given  series  is  -. 

4 

2.  Required  the  sum  of  the  infinite  series 

1.3^2.4^3.5^4.6^5.7'^ 
Let      .       Sr^l+l+l+l-^l  Sec,  ad  infin.  (1.) 

J>      O      't      0 

Or        -       S=       ?4.1+l-fl,  &c.,  a</«n/in.  (2.) 

2     3    4    5         '         -^  ^    ^ 

By  transposition, 

S-^=U1+^,  &c.,  ad  infin.  (3.) 

2     3     4     5 

Whence,  subtracting  the  last  equation  (3)  from  the  second 

(2),  we  shall  have 

2=^4- Ah- A,  &c.,  ad  infin. 
2     1.3     2.4     3.5'        '         -^ 

Or       -         ?=-L  +_L-}-_L,  &c.,  ad  infin. 
4     1.3     2.4^3.5'        '         -^ 

3 
Hence  the  sum  of  the  given  series  is  -. 

4 

3.  Required  the  sum  of  the  infinite  series 


2.4.6     4.6.8     6.8.10     8.10.12     10.12.14' 
III.  Third  Method. — The  following  method  may  some- 
times be  employed:  Assume  a  decreasing  series  containing 
26 


302  ELEMENTS    OF    ALGEBRA.  [sECT.  X, 

the  powers  of  a  variable  quantity  (a?),  whose  sum  is  equal  to 
S.  Multiply  both  members  of  this  equation  by  a  compound 
factor,  in  which  x  and  some  constant  quantity  are  contained; 
then  give  to  x  such  a  value  that  the  compound  factor  shall 
be  equal  to  0.  If  one  or  more  of  the  first  terms  be  then 
transposed,  these  will  be  equal  to  the  sum  of  the  remaining 
series. 

EXAMPLE. 

Let        .         S^l  +  ^+^'+^Vf-',  &c.,  ad  infin. 
2     3      4      5 


Multiplying  both  members  by  x — 1,  we  have 

-l^Z-^-Z.-Z.-'^,  &c.  ) 


S(a7— 1)_  >^  x_:^_x^_^_;^ 

'23456 


^  /y>  nn^  /)o3  /y^4  ^S 

Eed„ci„g,S(.-l)=-l  +  ^^+£-3+iL+?^+^^,&c. 

By  making  a;=:l,  the  equation  becomes* 

0=— !  +  —+—  +— +  —  +—,  &c. 
1.2     2.3     3.4    4.5     5.6' 

III.    RECURRING  SERIES. 

439.  A  recurring  series  is  one  which  is  so  constituted  that 
a  certain  number  of  contiguous  terms,  taken  in  any  part  of 
the  series,  have  a  given  relation  to  the  term  immediately 
succeeding.     Thus,  in  the  series 

l-^3x-\-4^x'-\-lx^-\-nx'+lSx\  &c., 
the  sum  of  the  coefficients  of  any  two  contiguous  terms  is 
equal  to  the  coefficient  of  the  following  term.     If  the  series 
be  expressed  by 

A+B-i-C+D+E+F,  &c.,  then 
The  1st  term        -         -         A=l  ; 
The  2d  term         -         -         B  =  3a?; 
The  3d  term        -        -         Cz=zBx-{-Ax^=4<x^ ; 
The  4th  term       -        -        I>  =  Cx-^Bx'=lx' i 


SECT.  X.]  RECURRING    SERIES.  303 

The  5th  term       -        -        E=Dx+Ca:«=lla?*  j 
The  6th  term       -        -        F=Ex-^Dj^=1Sx\  6cc. 
That  is,  each  of  the  terms  after  the  second  is  equal  to  the 
one  immediately  preceding   multiplied  by  x,  plus  the  one 
next  preceding  multiplied  by  x^.     Hence  all  the  terms  after 
the  first  two  are  subject  to  a  definite  law. 

440.  The  particular  expression  from  which  any  term  of 
the  series  may  be  found  when  the  preceding  terms  are 
known  is  called  the  scale  of  the  series^  and  that  from  which 
the  coefficients  may  be  formed  the  scale  of  the  coefficients. 

Recurring  series  are  divided  into  orders,  and  the  order  is 
estimated  by  the  number  of  terms  contained  in  the  scale. 

In  the  expansion  of  — —  in  Art.  436,  we  have  a  recurring 

series  of  the  first  order.     Thus, 

a         a     abx  ,  ah^oc^     aPx^    . 


c-^bx     c      cr* 

The  scale  of  the  coefficients  here  is  — _  j  that  of  the  terms 

c 

bx 
is  — — .     This  is  the  simplest  form  of  the  recurring  series. 

441,  In  a  recurring  series  of  the  second  order  the  law  of 
progression  depends  upon  two  terms,  and,  consequently,  the 
scale  consists  of  two  parts.  Let  m-{-n  represent  the 
scale  of  the  series,  and 

A  +  B-f  C  +  D-f  E+F,  &c., 
represent  the  recurring  series.     Then 

The  3d  term         -         -         C-Bmx-\-knx^ \ 
The  4th  term       -        -        'D  =  Qmx-\-Bnx^ ] 
The  5th  term       -         -         E  =  Dotx  +  C»x^  dec. 
Taking  the  last  two  terms  in  the  above  expression,  we 
have  the  two  equations 

D  =  Cmx+Bnx'  )  ^^  g^^  ^^^  ^^^^^^  ^^  ^  ^^^  ^ 
Ez^D^nx+Cwx'  S 
Since  the  scale  of  the  series  is  the  same,  whatever  be  the 
value  of  J?,  the  reduction  may  be  rendered  more  simple  by 
making  x=l.     The  equations  then  become 


304  ELEMENTS    OF   ALGEBRA.  [sECT.  X. 

E=:I>m+Cn, 
These,  reduced,  give 


DC-BE 


CE-DD 
CC-BD' 


CC— BD 

In  the  series  l  +  3a7+5j?'+7a?='+9j?*+lla7^  &c., 
A=:l,  B=:3x,  C^So^VD^Tar',  E=9a?^ 
Then,  making  x=.  1,  we  have 

7x5— 3x9_^2^         I        ^^5x9— 7x7^_.^ 


5x5—3x7  I  5x5-3x7 

44*2.  In  a  recurring  series  of  the  third  order  the  law  of 

progression  depends  upon  three  contiguous  terms.     Letting 

m-{-n-\-r.  represent  the  scale  of  series,  and 

A+B+C+D+E+F,  &c.,  the  series,  then 
The  4th  term       -         -         J)  —  Q>mx-\.Bnx^-\-krT^-, 
The  5th  term       -        -        '£,:=zDmx-\.Cnx'-\-Brx'' -, 
The  6th  term       -         -         Y-Emx-^J)nx'-\-Qrx\  &c. 
In  a  similar  manner,  we  may  obtain  the  succeeding  terms 

in  the  higher  orders  of  the  recurring  series. 

443.  To  ascertain  whether  th§  law  of  progression  depends 
on  two,  or  three,  or  more  terms,  we  may  first  make  trial  of 
two  terms ;  and  if  the  scale  of  the  series  thus  found  does 
not  correspond  with  the  series,  we  may  try  three  or  more 
terms.  If  we  begin  with  a  number  of  terms  greater  than  is 
necessary,  one  or  more  of  the  values  found  will  be  0,  and 
the  others  will  constitute  the  true  scale  of  valuation. 

444.  When  the  scale  of  a  decreasing  series  is  known,  the 
sum  of  the  terms  may  be  found. 

Let         -         a-^hx-\-cx'''-\-dx^-\-ex^-\-fx^^  &c., 
be  a  recurring  series,  whose  scale  of  relation  is  m-\-n. 
Then 


The  1st  term  -  -  =:A ; 

The  2d  term  -  -  =B  ; 

The  3d  term  -  -  C  =  'Bxmx-{-Axnx^ ; 

The  4th  term  -  -  'D  =  Cxmx-\-Bxnx'^ ; 

The  5th  term  -  -  E=:Dxwia?-f-Cx»a?^,  &c. 


SECT.  X.]  RECURRING    SERIES.  20Sk 

If  the  series  be  infinitely  extended,  the  la^t  term  may  be 
neglected  as  of  no  comparative  value  j  and  if  S=  the  sum 
of  the  terms,  we  shall  have 

S=rA-f  B  +  mxx(B  +  C  +  D,  &c.)  +  »ir*x(A-|-B-f  C,  &c.). 
But  B+C-f  D,  &c.»  =S— A,  and  A+B+C,  &c.,  =S. 
Hence,  by  substitution, 

Or         -         S=A+B+Smx— Amx+Sna?*. 

Transposing,  S  — Sotx — S«x^=A-f  B— Amx, 

Or         -         S{l—mx—nx')=A+E-Amx, 

T\.  .J.              o     A-fB — Amx 
Dividmsr,        o= — ! -. 


•&» 


1 — mx — nx^ 


EXAMPLES. 

1.  Required  the  sum  of  the  infinite  series 
l  +  6x+12xH48jc'+120a?*,  &c. 
A  =  l,  B=6j:,  C=12a:',  D=48ar',  E  =  120a;*,  &;c. 
Then,  making  x=l,  we  have 


^^12x48-6x120^.^ 
12x12—6x4:8 


^_  12x120-48x48    ^ 


12x12—6x48 
Substituting  the  values  of  A,  B,  wi,  and  n  in  the  formula, 
o_A  +  B — Amx 
1—vix—nx^^ 

We  shall  have     -     S=J"^^^""^. 
1 — X — 6ar 

Or       -        -        -     S=^±^. 
1 — X — 6a:' 

2.  Required  the  sum  of  the  infiMte  series 
l+2x-f  8x'+28a:'+100x*,  &c. 
Substituting,  as  before, 

^  =  a  ^^8x100—28x28 

8x8—2x28      ""'  "~ 


8—8—2x28 


q_l+2j— 3j__ 1- 


-X 


l-^Sx—^x"     1— 3x— 2x» 
3.  Required  the  sum  of  the  infinite  series 

l  +  3a:-}-4x'+7x'+llx*+18x'+29j?«,  &c. 

Qq 


306  ELEMENTS    OF    ALGEBRA.  [sECT.  X. 

4>.  Required  the  sum  of  the  infinite  series 

l-\-2x-\-3x^-}-^3c^+5x'+6x\  &c. 
5.  Required  the  sum  of  the  infinite  series 

l  +  3x-\-bx'-\-lx'-{-9x'-\-llx''-{-13x%  &c. 

IV.    METHOD  OF  DIFFERENCES.        ^     ' 

445.  We  will  now  proceed  to  point  out  another  process 
by  which  the  summation  of  various  kinds  of  series  to  a  lim- 
ited number  of  terms  may  be  obtained.  This  is  termed 
Method  of  Differences^  as  it  depends  on  finding  the  several 
orders  of  differences  belonging-  to  the  series. 

1.  Orders  of  Differences. 

1.  If  we  take  the  first  term  from  the  second,  the  second 
from  the  third,  the  third  from  the  fourth,  &c.,  in  the  given 
series,  the  remainders  will  form  a  new  series,  which  is  called 
the  first  order  of  differences. 

2.  If  we  proceed  with  this  new  series  in  the  same  manner 
as  with  the  given  series,  we  shall  obtain  the  second  order  of 
differences. 

3.  In  the  same  manner  we  may  obtain  the  third^  fourth, 
fifths  &c.,  orders  of  differences. 

446.  It  should  be  observed,  however,  that  when  the  sev- 
eral terms  of  the  series  increase,  the  differences  will  all  be 
positive  ;  but  when  they  decrease,  the  differences  will  be 
negative  and  positive  alternately. 

EXAMPLES. 

1.  Required  the  several  orders  of  differences  in  the  series 

1^  2^  3^  4^  b\  6^  &c. 
The  proposed  series         -         1,  4,  9,   16,  25,  36,  &c. 
1st  order  of  difference    -         -     3,  5,     7,     9,   11,  &c. 
2d  order  of  difference       -         -  2,     2,     2,     2,  &c. 

3d  order  of  difference      -         -         -        0,     0,     0,  &c. 

2.  Required  the  several  orders  of  differences  in  the  series 

1,  6,  20,  50,  105,  196,  &c. 
1st  order  of  difference  -  5,  14,  30,  55,  91,  &c. 
2d  order  of  difference     -         -      9,  16,     25,     36,  &c. 


SECT.  X.]  METHOD    OF    DIFFERENCES.  80f 

3d  order  of  difference  -  -  -.7,  9,  11,  A:c. 
4th  order  of  difference  -  -  -  -  2,  2,  <kc. 
3.  Required  the  several  orders  of  differences  in  the  series 

i»       h       i»       Vjt,       -Sly  &c- 

1st  order  of  difference    -     —\y  — J,  — ^,  — ^»j,  &c. 

2d  order  of  difference     -         -      -f  |,   -fjV,  +3^?  ^^* 

3d  order  of  difference     -         -         -     — J-^  — ^'^^  &;c. 

4th  order  of  difference    -        -        -        -        +3V>  ^c* 
2.  Law  of  the  Coefficients. 

44-7.  Letting  rr,  ft,  c,  rf,  &c.,  represent  a  series,  and  pro- 
ceeding with  this  series  in  the  same  manner  as  with  the 
preceding,  we  shall  likewise  obtain  the  several  orders  of 
differences. 

Proposed  series,      a,         ft,         c,        c/,        e,        /,  &c. 

1st  order  of  differ.,      ft — a,  c — ft,  d — c,  c — (/,  /— e,  &c. 

2d  differ.,  c— 2ft-|-a,  rf— 2c+ft,  A2rf+c, /— 26+</,  &c.  . 

3d  diff.,  (/— 3c+3ft — a,  e— 3(i4-'3c— ft,/— 3e-|-3</— c,  &c. 

4th  differ.,       e— 4</-}-6c— 4ft+a,/— 4e+6£/— 4c+ft,  &c. 

5th  difference  -  f—b€-{-\Od — 10c+5ft— a,  &c. 

448.  In  these  expressions,  each  difference  in  the  several 
orders,  whether  simple  or  compound,  is  called  a  term.  By- 
inspecting  the/r5^  terms  in  the  preceding  orders  of  differen- 
ces and  the  first  term  of  the  series,  we  shall  find  the  coeffi- 
cients to  be  as  follows  : 


1st  term  of  the  series 

1. 

Ist  order  of  difference 

-      1,     1. 

2d  order  of  difference 

-  h     2,     1. 

3d  order  of  difference 

1,     3,     3, 

4th  order  of  difference 

1,    4,     6,     4, 

5th  order  of  difference 

1,    5,    10,    10, 

1. 

5,    1; 

which  are  the  same  as  the  coefficients  in  the  powers  of  bi- 
nomials (Art.  265).  Therefore,  the  coefficients  of  the  first 
term  in  the  nth  order  of  differences  are  (Art.  429.) 

1,  n,  nx-^,nx_^x^- __X-^xn, 


308  ELEMENTS    OF   ALGEBRA.  [sECT.  X. 

3.  To  find  any  Term  in  the  SeYies. 
44<9.  In  order  to  obtain  a  general  expression  for  any  term 
of  the  series  «,  5,  c,  c?,  &c.,  when  the  differences  of  any  order 
become  at  last  equal  to  each  other,  let  d\  d\  d'",  &c.,  be  the 
first  terms  in  the  first,  second,  third,  &c.,  orders  of  differen- 
ces.     Then 

d'   =b — a; 
d"  =c—2b+a; 
c?'"=:6?-3c+36— a; 
d""=:e—U+6c—4>b-^a,  &c. 
Transposing^ and  reducing  these  several  equations,  we  ob- 
tain the  following  expressions  for  the  terms  of  the  original 
series : 

2d  term         -         hz=a-{-d' ; 
3d  term         -         c=a-\-'id'-\-d"  p 
4th  term        -         d=a'-\-W-\-W^d"' ; 
'  5th  term        -         e=za^^d'f6d"-\-4>d'"  +  d"'\  &c. 
450.  By  inspecting  the  above,  we  shall  discover  that  the 
coefficients  observe  the  same  law  as  in  the  powers  of  a  bi- 
nomial, with  this  difference,  that  the  coefficients  of  the  nth 
term  of  the  series  are  the  coefficients  of  the  {n—l)th.  power 
of  a  binomial.     Substituting,  then,  7i—l  for  n  in  the  formula 
for  the  coefficients  of  an  involved  binomial  (Art.  448),  and 
applying  the  coefficients  thus  obtained  to  d\  d'\  d'"^  d"",  «fec., 
as  in  the  preceding  equations,  we  have  the  following  gen- 
eral expression  for  the  nth  term  of  the  series,  a,  b,  c,  d,  &c. : 

T^th  term^a+^izl .  c/'+!^ .  !^  .  J"+^ZZ_1 .  !Ll^  .  ^11? . 
1  12  12         3 

d'^+^'szl .'^^.'Lrl.'LrA. d'"'  &c. 

12  3         4 

JN'ote. — When  the  differences,  after  a  few  of  the  first  or- 
ders, become  0,  any  term  of  the  series  is  easily  found. 

EXAMPLES. 

1.  Eequired  the  12th  term  of  the  series  2,  6,  12,  20,  30, 

&c. 
Proposed  series    -        .        .        2,  6,,   12,  20,  30,.  &c. 


SECT.  X.]  METHOD    OF    DIFFERENCES.  309 

Ist  or-der  of  difference  -  -  -  4,  6,  8,  10,  &c. 
2d  order  of  difference  -  -  -  -  2,  2,  2,  &c. 
3d  order  of  difference  -        -        -         -  0,     0,  &c. 

Here  </'=-4.,  (/"=2,  and  n=12;  and  as  d'"=0,  it  will  be 
necessary  to  use  only  the  first  three  terms  of  the  formula. 

Hence,a+!^.rf'+!?=li.!L:?.c/"=2+l?i:i.4+i^Z:-^ 
112  11 

l^II?.  2=2+44  +  110=156.  ^ns.  12th  term  =156. 

2.  Required  the  20th  term  of  the  series  1,  S,  27,  64-,  125, 
&;c. 


1,  8,  27,  64.,  125,  &c. 

-  7,  19,  37,     61,  &c. 

-  12,  18,     24.,  &c. 

6,       6,  &c. 
0,  &c. 
6,  and  d""=0  ;  therefore 


Proposed  series  - 
1st  order  of  difl'erence  - 
2d  order  of  difference  - 
3d  order  of  difference  - 
4th  order  of  difference  - 
Here  n=20,  rf'=7,  d"=12,  d' 
only  four  terms  of  the  formula  will  be  required. 

Hence,  a+!^  .  rf'+!Lll .  !L-Z?  .  cf"+?^l .  !Ll^  .  !LI?  . 
'  1  1         2  1         2         3 

^,/^,  ,  20— 1   7,  20— 1   20—2   JO     20— 1   20—2   20—3 

l''!"l'2*"l'2'3' 

6=1  +  133  +  2052+5814=8000.     ^ns.  20th  term  =8000. 

3.  Required  the  15th  term  of  the  series  1,  4,  9,  16,  25,  36, 

&c.  »^ns.  255. 

4.  Required  the  50th  term  of  the  series  1,  3,  6,  10,  15,  21, 

&c.  Jins.  1275. 

5.  Required  the  30th  term  of  the  series  1,  |,  ^,  i^y,  j^j,  ^Vi 

&c.  JJns,  j}j. 

4.  To  find  the  Sum  of  n  Terms. 
In  order  to  find  the  sum  of  n  terms  of  the  series  a,  ^,  c,  d, 
&c.,  when  the  differences  of  any  order  become  at  last  equal 
to  each  other,  let  one,  two,  three,  &c.,  terms  be  successive- 
ly added  together,  so  as  to  form  a  new  series,  as 
0,  a,  a  +  6,  a-\-b-^Cy  a  +  6  +  c  +  c/,  &;c. 
Taking  the  diffferences  in  this  series,  we  have 


310  *  ELEMENTS    OF    ALGEBRA.  [sECT.  X. 

1st  difference       -         -        -     «,        ^)         c,        «?,  &c. 

2d  difference        -         -         h — a,  c— 3,  d — c,  c— /,  &c. 

3ddiiier.,    c— 26+a,  t/— 2c+^>,  e— 2£;+c,/-2e+c/,  &c. 

4thdiff.,(^— 3c+36— a,e— 3c/+3c— Z*,/— 3e+3ri— c,  &c. 

Here  it  will  be  observed  that  the  first  order  of  differences 
in  the  new  series  is  the  same  as  the  original  series,  and  the 
second  order  of  differences  is  the  same  as  the  first  order  in 
the  original  series  a,  6,  c,  d^  &c.  j  and,  generally,  that  the 
(n-{-\)i\i  order  in  the  new  series  i^  the  same  as  the  nih.  order 
in  the  original  series. 

In  this  case, 
0    =  1st  term;  a    =  1st  order  of  difference ; 

d'  =:2d  order  of  difference  ;     d"  =:3d  order  of  difference  ; 
d"'z=iA>\\i  order  of  difference  ;  6?""=: 5th  order  of  difference. 

Resuming  now  the  formula  (Art.  450) 
,  n — 1     1,  ,  n — 1    n — 2    ,,-  ,  n — 1  \ — 2   n — 3     ,,,,    „ 

^1  1  2  1  2  3  '         ' 

which  is  the  general  expression  for  the  n\\v  term  of  a  series 
whose  first  term  is  a  ;  applying  it  to  the  new  series,  in  which 
the  first  term  is  0,  and  substituting  n-\-l  for  n,we  have 
n  ,        ,  n    n — 1      1,  ,  n    n — 1    n — 2    j.,  ,  n     n — 1     n — 2 
^12  12  3  12  3 

__  .  d  ,  &c., 

4 

^  which  is  a  general  expression  for  the  (w  +  l)th  term  of  the 
series 

0,  a,  a+J,  a+Z>+c,  a+o+c+t^,  &:c. ; 
Or  the  Tith  term  of  the  series 

a,  a-\-h^  a+^+c,  a-\-h-\-c-\-d^  &c. 
But  the  n\\v  term  of  the  latter  series  is  evidently  the  sum  of 
n  terms  of  the  series  a,  Z>,  c,  d^  &;c. 

Hence,  the  general  formula  for  the  sum  of  n  terms,  a  series 
of  which  a  is  the  first  term,  is 

,  n   n — 1     7,  ,  n   n — 1    n — 2     »,,  ,  n   n — 1    n — 2    n — 3 

na-\--  . d  +-  . . .  a  +-  •  • • • • 

^12  1,  ,2  3  ^12  3  4 

d"',  &c. 


SECT.!.]  METHOD    OF    DIFFERENCES.  311 

EXAMPLES. 

1.  Required  the  sum  of  n  terms  of  the  series  1,  2,  3,  4, 

5,  6,  &c. 
Proposed  series       -         -         -       1,  2,  3,  4,  5,  6,  &c. 
1st  order  of  difference     -         -         -  1,   1,  1,   1,   1,  &c. 
2d  order  of  difference      -         -         -        0,  0,  0,  0,  &c. 
Here  a=l,  (/'=1,  and  d=zO  j  therefore, 

na+ .(f=n-\--  . =n+ =  — I— =  sum  of  n 

12  12  2  2 

terms. 

In  the  above  example  let  n=20  j  then 

n^^400-20^210.     ^ns. 
2  2 

2.  Required  the  nth  term  of  the  series  of  odd  numbers 

1,  3,  5,  7,  9,  4:c. 

Proposed  series       -         -  -         -  1,  3,  5,  7,  9,  &c. 

Ist  order  of  difference     -  -         -       2,  2,  2,  2,  «fec. 

2d  order  of  diflJerence      -  -         -         -    0,  0,  0,  &c. 

Here  a=l,  t/'=2,  and  d'=0  j  therefore, 

na-\--  . .  a  =  -\- .  2=n*. 

12  2 

Hence,  Me  sum  of  the  terms  is  equal  to  the  square  of  the 

number  of  terms.  ' 

3.  Required  the  sum  of  n  terms  of  the  series  P,  2',  3*,  4*, 

5^  6',  r,  &c.,  or  1,  4,  9,  16,  25,  36,  49,  &c.  Also, 

the  sum  of  20  terms. 

Proposed  series       -         -  1,  4,  9,   16,  25,  36,  49,  &c. 

1st  order  of  diflference      -        3,  5,     7,     9,   11,  13,  <fec. 

2d  order  of  diflference     -         -    2,     2,     2,     2,     2,  &c. 

3d  order  of  difference      -         -  0,     0,     0,     0,  &c. 

Here  0=1,  (/'  =  3,  d"  =  2,  and  d'=Q  ;  therefore, 

12  12  3  2 

2»»-6n'+4»_6»  ,  9n'-9n  .  2n'-6n'+*»_l  /o»«_L 
6 T"^— 6— + ^6— 6"^      ^ 

3»-f- 1)=  the  sum  of  n  terms. 


312  ELEMENTS    OP    ALGEBRA.  [sECT.  X. 

Or,if7i=:20,-?^(27^'+3;^-hl)=:-.20(2.20^+3.20+l)  =  2870. 
6  6 

4.  Required  the  sum  of  n^  and  also  of  50  terms  of  the  se- 

ries 1^  2^  3',  4',  5^  &c. 

Jins.  n  terms = -, — ! — 

50  terms=  1625625. 

5.  Required  the  sum  of  n,  and  also  of  12  terms  of  the  se- 

ries 1^  2\V,  4^  5^  6*,  &;c. 

Ans.  n  terms=^V-+-— — ^ 
5      2      3     30 

12  terms=60710. 

6.  Required  the  sum  of  w  terms  of  the  series  1^,  2^,  3^,  4\ 

5^,  &c. 

6      2      12      12 

7.  Required  the  sum  of  n  terms  of  the  series  2,  6,  12,  20, 

30,  &c. 

^;^..7^terms=<^+iK!^±^. 
3 

8.  Required  the  sum  of  w,  and  also  of  20  terms  of  the  se- 

ries 1,  3,  6,  10,  15,  &c. 

Ans,  n  terms=<^±lH^±l). 
1.2.3 

20  terms=1540. 

9.  Required  the  sum  of  n  terms  of  the  series  1,  4-,  10,  20, 

35,  &c. 

Ans   <^^l)(^+2)(7z+3) 
1.2.3.4 

V.    REVERSION  OF  SERIES. 

451.  To  revert  a  series  is  to  express  the  value  of  the  un- 
known quantity  in  it  by  means  of  another  series  involving 
the  powers  of  some  other  quantity. 

Let  X  and  y  represent  two  indeterminate  quantities,  and 
let  the  value  of  y  be  expressed  by  a  series  composed  of  the 
powers  of  x  ;  thus, 

y—ax-\-lx^-\-cx^-\-dx'^-\-^  &c.. 


SECT.  X.] 


REVERSION    OF    SERIES. 


313 


in  which  a,  by  c,  J,  d:c.,  are  known  quantities  ;  then,  to  re- 
vert  the  series  is  to  express  the  value  of  a*  in  a  series  con- 
taining only  y,  and  the  known  quantities  a,  6,  c,  (/,  <fcc. 

Then,  in  order  to  express  the  value  of  x  in  terms  of  y,  as- 
sume x=Ay-f  By'+Cy'+Dy^H-,  &c. 

Substituting  this  value  for  x  in  the  proposed  series,  and 
transposing  y,  it  will  become 


0=aA 
—1 

Whence  w 

y-haB 
+6A« 

e  have 

+  2^»AB 
+cA» 

-f26AC 
+iB' 
+  3cA'B 

+dA* 

y*+,&c. 

aA— 1     -         - 

=  0,  andA=i; 
a 

aB+JA* 

=0,  andB=-*; 

aC+26AB  +  cA*      - 

.         -0,andC-2*'-«i 
a 

aD+2*AC-f 

&B^4.3cA^ 

B+(/A*=0, 

«n^  n__5i-5«Jc4-a«d 

&c. 

Substituting  these  values  of  A,  B,  C,  D,  &c.,  we  have 


1  *  2    . 

«=-xy—-xy'-}- 


2Z>' 


■ac 


xy*- 


56'-5ffic-f-a*c/ 


x/4-,&c.  (1) 


a  cr  a- 

If  the  series  be  of  the  form 

y=<ix-\-bxr^-\-cx^-{-,  <Scc., 
in  which  the  even  powers  of  x  are  not  contained,  then  we 
shall  obtain  instead  of  the  above  formula    (1)' 


1  b       .  ,  W—ac       . 


^±^z:«f*£xy-Ac.(2, 


EXAMPLES. 

1.  It  is  required  to  revert  the  series  y=x-f-a:*+ar*-|-,  dec. 
Here  a=l,  6=1,  and  c=\  ;  therefore. 


=  1, 


.A=-l,5^«f=l,  and-^^!i:!±±^=-l. 
a* 


a  <r  a* 

Hence  a:=y — y'+y* — y*,  &c. 
27  Rr 


314  ELEMENTS    OF    ALGEBRA.  [SECT.  XI. 

2.  Revert  the  series  x=:2y-{-Sy^-\-4y'^-\-5y''  +  ,  &c. 
Here  a=2^  b=3,  cr=4,  dz=zd,  &c.  5  therefore, 

a~2'      a^~     16'  ~~a'  128'  ^^  a^° 

152 


1024 


Hence  — y=ir— Aa?'+iia?^_i5?.a:'  &c. 
^     2       16        128        1024 

3.  Revert  the  series  y^x — -o^-\--o^ — -a?'*+,  &;c. 

^  2        4        8         ' 

Ans.  a7=2/+V+ V+ V+,  &c. 

/^         4         o 

4.  Revert  the  series  v=a: — -x^-\--x^ — -a?^  &c. 

^  3        5        7    ' 


SECTION  XL 

GENERAL  THEORY  OF  EaUATIONS. 

General  Properties  of  Equations. — Composition  of  Equations. — 
Transformation  of  Equations. 

GENERAL  PROPERTIES  OF  EQUATIONS. 

452.  Every  complete  equation  of  the  nth  degree,  n  being 
a  positive  whole  number,  if  it  involves  but  one  unknown 
quantity,  may  by  reduction  be  put  under  the  form 
Aa:"+Ba?"-'+Ca;"-24- +Ta:+U=0. 

If  this  equation  be  divided  by  A,  and  the  coefficients  -,— , 

T         U 
....—,  and  —  be  represented  by  ^,  c,  .  .  .  t,  and  u,  we  shall 

A  A 

have 

a^"_|_Ja^"-i_j_cj7"-2+ -\-tx-\-u^O. 


SECT.  XI  ]         GENERAL   THEORY    OF   EQUATIONS.  315 

^ote. — Any  real  or  imaginary  algebraic  expression  which, 
being  substituted  for  x  in  an  equation,  satisfies  its  condi- 
tions, is  called  a  root  of  that  equation. 

453.  Theorem. — If  a  represent  any  root  of  the  equation 

x^_|_Jx^'-j-cx"~'4- -\-tx-\-n=Oy  the  first  member  of 

this  equation  is  divisible  by  x — a. 

Demonstration. — By  supposition,  x=a  ;  then,  substituting 
a  for  X,  we  have 

a"+6a'^*-f  ca»-'+ -\-ta-\-Uz=0. 

Or,  by  transposition, 

u= — a" — Aa**"' — ca"^' — — ta. 

Substituting  this  value  for  u  in  the  original  equation,  we 
have 

x"4-ix*-'+cx*^'+ -\-tx  >  _Q 

_^---fta«-»>_ca"-*— —ta  S 

Or,  by  uniting  the  corresponding  terms,  the  equation  be- 
comes 
(x"— a'')-hi(a:^'— a'-')+c(a:*-«— a''-»)-h  .  .  +t(x—a)=zO. 
In  this  equation 


X 


n— 1        „n—l 


a>n-2_^n_«^ 


X     — a, 
are  each  divisible  by  a?— a  (Art.  203,  th.  7)  j  therefore,  the 
first  member  of  the  original  equation  is  also  divisible  by 
X — a. 


EXAMPLES. 


Suppose  2  to  be  a  root  of  the  equation 

a:*— 16x^4-56  =  0. 
By  the  theorem  just  demonstrated,  the  first  member  of  this 
equation  must  be  divisible  by  x— 2.     Thus, 


olG  ELEMENTS    OF  ALGEBRA.  [^ECT.  XI. 

gg— I6ar^4-56|a?— 2 

a?3_  2x^         x^—Ux-2a 


— 14a;24-28a? 


—28074-56 
—280? +56 

0 

Corollary  1.  If  we  divide  the  general  equation 

a7"  +  5a?"-'  +  ca;"-'+  .  .  .  -^tx-\-u=0,  (1.) 

by  a; — a,  there  will  result  a  new  equation  one  degree  less 
than  the  given  equation,  which  may  be  put  under  the  gen- 
eral form 

a?"-'+Z»V-2+c'a7"-^+,  &c.=iO.  ,  (2.) 

Hence,  the  original  equation  may  be  transformed  into  the 

following  equivalent  expression  : 

(a?— a)(a?'^-'  +  6V-2+c'a?"-''+,  &c.)=0. 

The  conditions  of  this  equation  are  satisfied  on  the  sup-'' 

position  that 

x  —  a. 

Cor.  2.  The  result  (2)  obtained  in  the  preceding  corollary 
may  evidently  be  divided  by  x — a',  if  a^  represent  a  root  of 
that  equation  5  then, 

x^-'-\-b"x^+c"x^-^-{-,  6lc.,=0.  (3.) 

Hence  we  shall  have 
a:n-i4.2,'a?"-'+cV-^+,  &c.  =  (x—a)  (a?"-^  +  fx""^  +  c"a?"-*+, 

&C.)=::0, 

and  the  original  equation  becomes 

(x—a)  (x—a')  (a:"-2+^"ar-«+c"ic^+,  &c.)=0. 
The  conditions  of  this  equation  are  satisfied  on  either  of 
the  following  suppositions,  viz. : 

x=a, 
Or      -         -         -         -      x=za\ 

Proceeding  in  the  same  way  to  find  the  remaining  roots, 
a",  a'",  &c.,  the  original  equation  will  eventually  assume  the 
form 

(x-a)  (x—a)  (x—a")  (x—a'")  (x—a""),  &c.,=0.    (4.) 


SECT.  XI.]         GENERAL   THEORY    OF    EQUATIONS.  317 

Cor.  3.  In  the  above  (4)  equation  there  are  evidently  n 
factors.  Hence,  the  number  of  roots  of  an  equation  is  denoted 
by  the  degree  of  the  equation.     Thus, 

An  equation  of  the  second  degree  has  tioo  roots  ; 

An  equation  of  the  third  degree  has  three  roots  ; 

An  equation  of  the  fourth  degree  has  four  roots^  &c. 

Scholium.  If  any  of  the  factors  into  which  the  first  mem- . 
ber  of  the  equation  may  be  resolved  are  equal,  the  number 
of  unequal  roots  will  evidently  be  less  than  the  number  of 
units  in  the  exponent  expressing  their  degree. 

EXAMPLE. 

The  equation  (x— a)*  {x—a'f  (x — a")'  (x — a")=0  has  but 
four  different  roots,  although  it  is  an  equation  of  the  10th 
degree. 

Cor.  4.  If  one  root  of  a  cubic  equation  be  found,  and  the 
equation  be  divided  by  the  simple  equation  containing  that 
root,  the  quotient  will  be  an  equation  of  the  second  degree 
containing  the  other  roots. 

EXAMPLES. 

1  If  one  root  of  the  cubic  equation  a^ — 7x'+36=0  is  3, 
what  are  the  other  two  roots  1 

By  the  conditions  of  the  problem,  x=3  .*.  x— 3=0. 
Dividing  the  given  equation  by  this,  we  have 

X*— Tx'-h  36|x— 3 

x*— 3x«  x'— 4x— 12 


—4x^4-36 
— 4ir'-|-12x 

— 12XH-36 
— 12X+36 

0 
Hence  we  have  the  quadratic  equation 
x"— 4x— 12:r0, 
which,  reduced,  gives  x=6,  or  — 2,  the  other  two  roots  of 
the  cubic  equation.     Hence,  the  three  roots  of  the  proposed 
equation  are  3,  6,  and  — 2. 


318  ELEMENTS    OF    ALGEBRA.  [sECT.  XI. 

2.  If  one  root  of  the  equation  x^-{-ixr — 16a;-f  20  =  0,  is  — 5, 
what  are  the  other  two  roots  ]  ^ns.  2  and  2. 

3.  If  one  root  of  the  equation  x^-\-Zx'' — 10y=0,  is  2,  what 
are  the  other  two  roots  ]  Ans.  — 5  and  0. 

CoR.  5.  If  two  roots  of  an  equation  of  the  fourth  degree 
be  given,  the  remaining  two  njay  also  be  found  ,•  and  so  of 
the  higher  equations. 

EXAMPLE. 

1.  Two  roots  of  the  equation  a;*--3a;^— 1437^4-4807—32  =  0, 
are  1  and  2  ;  what  are  the  other  roots  1      *Mns.  4  and  — 4. 

CoR.  6.  Equations  of  the  form 
a;"=a 
would  appear  to  have  but  one  root  j  but,  from  the  preceding 
reasoning,  it  must  have  n  roots. 

EXAMPLES. 

1.  What  are  the  two  foots  of  the  equation  a?^=4  1 

Ans,  2  and  —2. 

2.  What  are  the  roots  of  the  equation  x^—l  % 
Extracting  the  cube  root,  we  obtain 

a:=l. 

Consequently,  1  is  one  of  the  roots  j  then,  to  ascertain  if  it 

has  any  more  roots,  we  may  put  the  equation  under  the  form 

0?^— 1=0. 

This  equation  must  be  divisible  by  x — 1  ;  therefore, 

3?='— l  =  (a?— 1)  (a?2+a?+l)=0, 

Or        -        -        -        -        x'+x+l  =0.         

The  roots  of  this  last  equation  are  |( — 1-\-\/ — 3),  and 
|( — 1 — ^ — 3).  Hence,  the  three  roots  of  the  equation 
a?'=l  are  1,  ^(— I+n/"^),  and  |(— 1— x/^^3). 

3.  What  are  the  roots  of  the  equation  37"  =  !  1 

Ans.  1,-1,  s/~—i,  and  —^'—i, 

4.  What  are  the  roots  of  the  equation  aj^=l  1 

Ans.  1,  andi(^— 1— v/5±  y/— 10±2v^5J. 

COMPOSITION   OF  EQUATIONS. 

454.  From  what  has  been  said,  it  will  be  readily  inferred 


SECT.  XI.]    GENERAL  THEORY  OF  EQUATIONS. 


319 


that  equations  of  any  degree  higher  than  the  first  may  be 
produced  by  the  successive  multiplication  of  equations  of 
the  first  degree. 

Let    -        -        -      x— 2=0, 

And    -        -        -      X— 3=0: 


X— 4         =0, 

ar*— 9x^+26a?- 
X — 5 


-24=0. 
=  0, 


Multiplying 
Again,  let  - 

Multiplying 
Again,  let  - 

Multiplying         -      a:*— 14x»+71a:«--154a:-f-120=0,  &c. 

Hence,  the  product  of  two  equations  of  the  first  degree  is  an 
equation  of  the  second  degree  ;  the  product  of  three  equations  of 
the  first  degree  is  an  equation  of  the  third* degree,  &c. 

The  above  equation  of  the  fourth  degree  has  evidently 
four  roots,  viz.,  2,  3,  4,  and  5. 

455.  The  law  by  which  the  coefficients  are  governed  may 
be  seen  by  inspecting  the  results  obtained  by  the  actual 
multiplication  of  the  factors. 

Let  a,  a\  a",  a'",  a"'\  &c.,  represent  the  roots  of  the  gen- 
eral equation 

a?'*+6x'^'+ca:''-2+,  &c.,  =  0. 

Then  we  shall  have  (by  Art.  453,  Cor.  2) 
x^-f- Jj:'^'  +  cjr-'+,  &c.,=(x— a)  {x—d)  (x—a")  (x— a"0,&c., 

=0. 
Or,  multiplying  the  factors  and  writing  the  coefficients  of 
the  same  power  of  x  under  each  other,  we  have 
1.  [x — o)  (* — a')      -        -      =x' — a    la:  +aa'=0. 


:0. 


T-ffla'a"tt"'=0 


2.(jr-a)(x-a')(x-a") 


3.  [x—a)  (r— «')  (r— a")  (x— a'")=a 


—a' 

—a 

xHoa' 

x—aa'a": 

—a' 

+aa" 

— a" 

+a'a" 

*-a 

T^-aa' 

x^aa'a'' 

— a' 

-\-aa" 

—aa'a"' 

—a" 

+aa'" 

-aa"a" 

-a'" 

+a'a" 

—a'a"a' 

+aV" 

•^a"a"' 

320  ELEMENTS    OF   ALGEBRA.  [SECT.  XI. 

456.  By  attending  carefully  to  the  above  results,  we  shall 
discover  the  following  properties  : 

1.  The  coefficient  of  x  in  the  first  term  is  always  1. 

2.  The  coefficient  of  x  in  the  second  term  is  the  sum  of  all  the 
roots  of  the  equation  taken  with  contrary  signs. 

Thus,  the  roots  of  the  equation  of  the  second  degree  are 
a  and  a';  the  coefficients  of  x  in  the  second  term  are  — a, 
and  — a\  In  the  cubic  equation  the  roots  are  a,  a\  and  a"; 
the  coefficients  are  — a,  — a\  and  — a".  In  the  equation  of 
the  fourth  degree  the  roots  are  a,  a\  a'\  and  a'";  the  coef- 
ficients are  — a,  — a\  — a",  and  — a"', 

3.  The  coefficient  of  x  in  the  third  term  is  the  sum  of  all  the 
products  of  the  roots  taken  two  and  two,  and  so  on. 

Thus,  in  the  equation  of  the  fourth  degree,  the-roots  are 
a,  a  J  a",  and  a";  and  the  coefficients  in  the  third  term  are 
aa\  aa"y  aa"\  a'a[\  a'a"\  a" a'". 

4i  The  last  term,  which  is  independent  of  x,  is  the  product 
formed  from  all  the  roots  of  the  equation  after  the  signs  are 
changed. 

Thus,  in  the  cubic  equation,  the  last  term  — aaa'= — ax 
— a' X — a"  ;  and  in  the  biquadratic  equation,  the  last  term 
-\-aaa"a"'—-^a  X  — a'  X  — a"  x  — a"\ 

CoR.  1.  If  the  roots  are  all  negative,  the  terms  of  the  equa- 
tion to  which  they  belong  will  all  be  positive. 

For,  letting  -  -  x——a, 
x^—a" 
xz= — a'\  &c. 

By  transposition,  we  have  a;+a  =  0,  a;+a'=0,  a?4-a"=r0,  &c. 

Consequently, 

{x-^a)  (x-\-a)  {x-\-a")',  &c.,  =  0. 

CoR.  2.  If  part  of  the  roots  are  positive  and  part  negative, 
part  of  the  terms  of  the  equation  to  which  they  belong  will 
be  positive  and  part  negative. 

TRANSFORMATION  OF  EQUATIONS. 

457.  The  transformation  of  an  equation  consists  in  chan- 
ging its  form  without  destroying  the  equality  of  its  members. 


SECT.  XI.]         GENERAL   THEORY    OF    EQUATIONS.  321 

458.  TuEOREM. — *dny  proposed  equation  may  be  transformed 
into  another,  the  roots  of  which  shall  be  any  multiples  or  sub' 
multiples  of  those  of  the  former. 

First,  In  order  to  demonstrate  the  above,  let  us  resume 
the  general  equation 

x"+^'-'+cx"-»+ +^x+tt=0. 

Let  y  represent  the  unknown  quantity  of  a  new  equation, 
of  which  the  roots  are  a  times  greater  than  those  of  the  pro- 
posed equation ;  then 

y=ax,  and  x=±. 
^         '  a 

Substituting  this  value  for  x  in  the  general  equation, 
y-A^b^     ^c^ U _l/Llw=0. 

Multiplying  by  a", 

y"4-%'-'+ca'y''--+ +/a'*-'y+a''tt=0.         (1.) 

This  last  equation  will  evidently  fulfil  the  conditions  re- 
quired, since  y=ax 

Secondly,  Lety=:-;  then  a: = ay. 
a 

Substituting  and  reducing,  as  before,  we  shall  obtain 

V'+ly'-'+~y-'+ +^y+^=o-     (2-) 

Corollary.  Since  the  coefficients  in  the  preceding  equa- 
tion (1)  are  multiples  of  the  coefficients  in  the  general  equa- 
tion, it  is  evident  that  any  equation  having  fractional  coeffi- 
cients may  be  transformed  into  another,  in  which  all  the 
terms  shall  be  entire  numbers,  and  the  coefficient  of  whose 
first  term  shall  be  unity. 

EXAMPLE. 

Transform  the  equation  ar'-fir^-|-|a?+J=0. 
Multiplying  this  equation  by  12,  the  least  common  multi- 
ple of  the  denominators, 

l2x''\-Gx'-\'Sx-\-9=0, 
In  this  equation  all  the  terms  are  entire  numbers,  but  the 
coefficient  of  the  first  terra  is  greater  than  unity. 

Ss 


322 


ELEMENTS    OF   ALGEBRA. 


[sect.  XI. 


Then,  let 


y=12a;,  and  a?=J^. 
^  '  12 


Whence,  by  substitution, 

122^12^^12  ^ 
Multiplying  by  12^,     2/='4-6/+96y+1296  =  0. 
If  the  value  of  y  in  this  equation  be  found,  that  of  x  can 

be  readily  obtained,  since  x=^. 

459.  Theorem. — »dn  equation  may  ie  transformed  into  an- 
other^ the  roots  of  which  shall  be  greater  or  less  than  those  of  the 
former  by  a  given  number. 

Let  us  resume  the  general  equation 

a:"+Z'a:"~'+ca;"~2+  ....  ■^tx-\-u=0^ 
and  suppose  it  were  required  to  transform  it  into  another, 
whose  roots  (y)  shall  be  less  or  greater  than  those  of  the 
given  equation  by  e. 

First.  Let  x=y-^e. 

By  substituting  y+e  for  x  in  the  general  equation,  we  shall 
obtain 

(y+e)"+i(y+e)'-'+c(y+e)'-2+ +^(y+e)+i*=0. 

Or,  expanding, 


-Vb 


^  2 

-\-{n-\)be 
+  c 


y"-2+ +e" 


0.    (L) 


■\te 

This  equation  will  evidently  fulfil  the  conditions  required, 
since  y—X—e. 

Secondly.  Let  x^=:y—e  ;  then. 
Substituting,  as  before,  we  have 

{y—ey-\-b{y—e)"-'-\-c(y—ef-''-\- -\-t[y—e)-\-u  =  0. 

Or,  expanding. 


8BCT.  XI.]   GENERAL  THEORY  OF  EQUATIONS. 


323 


y* — n^ 


{n 


I)?.' 


3^-'+,  <kc.,  =0.        (2.) 


— (n— l)Je 
+  c 

Which  equation  also  fulfils  the  required  conditions,  since 
y=x^e. 

Cor.  1.  Since  the  truth  demonstrated  in  the  above  theo- 
rem  does  not  depend  upon  any  particular  value  of  e,  or  since 
e  is  indeterminate,  it  may  be  taken  to  satisfy  any  proposed 
condition.  Letting  it,  then,  be  taken  of  such  value  that  the 
coefficient  of  y'*~'  may  be  equal  to  zero,  or  — we-f6=0,  in  the 
above  equation  (1),  the  second  term  vanishes,  and  the  equa- 
tion becomes 


^  2 


y"-»+,  d:c.,  =0. 


(3.) 


+  c 

Or,  representing  the  coefficients  by  c',  d\  &c.,  we  shall  have 

y"-l-c'y'^+</y'^+ -\-u=0. 

This  condition  is  represented  by  the  equation 


which  gives 


h 
e=  — ; 

n 


whence  the  second  term  of  an  equation  may  be  removed  by  sub- 
stituting for  the  unknown  quantity  some  other  unknown  quanti' 
ty,  together  with  such  a  part  of  the  coefficient  of  the  second  term, 
taken  with  a  contrary  sign,  as  is  denoted  by  the  index  of  the 
highest  power  of  the  equation, 

EXAMPLE. 

Transform  the  equation  r* — 9a^-\-lx-\-l2=0  into  one  which 
shall  want  the  second  term. 

Let        -        -        -       x=y  +3 ; 

Then  -  -  -  «»=y'H-9y^+27y+27 
—dx'^  -9y'-54y-81 
+7x  =  +  7y+21 

+12  =  +12 


=0. 


Whence  x^— 9a:«+7a+12  =y»         -20y-21  =0. 


324  EL-EMENTS    OF   ALGEBRA.  [sECT.  XI. 

Cor.  2.  In  the  same  manner,  the  third  term  may  be  remo- 
ved from  an  equation  by  taking  the  value  of  e,  such  that 

^{^-'^Je'-(n-l)he+c=.0. 

For,  transposing  c,  and  dividing  by  the  coefficient  of  e',  we 

obtain 

,    2J  2c 

e — 6: 


n  n{n — 1) 


Reducmg        -        -       «=-=t\/— zr— -^+^2* 

n      y       n{n — 1)     n 

EXABIPLE. 

Transform  the  equation  x^—6x'^-\-9x—l=0  into  one  in 

which  the  third  term  shall  be  wanting. 

„  6_L.     /       2x9     ,36     n_u     /     18  ,36     ^  . 

Here    e^-±  ^ -^^^^^  + -=2±  y/ ^-  +  -=2± 

^_3-t-4=3,  or  1. 
Then,  letting      -       x  =y -\-S, 
we  shall  have         -       x'=.y'-{-9f-\-27yi-2'7  ] 
-6^2=     _6/-36y-54   i 
-\-9x=  +  9y4-27  (  ~^' 

-1    =  -Ij 


Whence  x'—6x^-i-9x—l=f-^3y^  —   1=0. 

Or,  taking  e=l,  and  proceeding  in  the  same  manner,  we 
shall  obtain  y^--3y-\-3=i0. 

CoR.  3.  Since  n,  in  the  general  equation,  is  indeterminate, 
an  equation  of  any  degree  may  be  transformed  into  another 
from  which  the  second  or  third  term  shall  be  removed. 

EXAMPLE. 

Transform  the  equation  x^ — Sx^-\-ox^ — 10a7+4=0  into  an- 
other that  shall  want  the  second  term. 


Let         -         x=y+l=y+2. 
n 

Then     -         a?^ =3^^+82/'+ 24^^+32^+16  ^ 

— 8x'=    _8/— 48/— 96y— 64 

+  5a^'=             +  5/+20y+20 

— lOo?  =                          _10y— 20 

+  4      =:                                                 +    4   , 

Whence                y"         —  i9/_54y_44_ 

:0. 

=0. 


SECT.  XI.]         GENERAL    THEORY    OF    EQUATIONS.  326 

Cor.  4.  If,  in  the  general  equation, 

a:"-|-6x'*-'  +  cx'*^^+ +^a:-|-u=0, 

we  let  x=y+e,  and  this  value  be  substituted,  the  equation 

becomes 

(y^.g)*^6(y+c)'-'+c(y-f  0*^ s{y+ey-\-t{y-{-e)-\-u=0. 

Expanding  each  binomial  term  of  this  equation  separately 

(1).  (y+«r=y"+„3r'.+fcl)3r^«' ....  "'"-^'  ] 

y'c'^«+ny€"-'+e" 

(2).  b{y  +  e)-'  =  by'*-'  -f  (»— 1)  hy^'e  ^(^riHi^II?) 

l\f^^(?  ....  (n—\)bye'^+le^'      - 
(3).  c(y  +  e)'*-'  =  cjr^  +  (;i— 2)  cjT'e  +  (^H^K^H^) 

cy^-^c*  ....  ce"^' ^  =0. 

(4).  d(y+ey^  =  c/y^  +  (n-3)(/y^e  +  (^— 3)  (>»— ^) 
d'y'^-'e' 

(n— 2).  r(y+e)'=ry'H-3ry'e+3rye'+re* 

(n— 1).  *(y4-e)'=^*+2^«  +s^ 

(»).         ^(y+c)  =/y  4-^e     -         -        - 

(n-4-1).        w     =M 

By  inspecting  the  above  results,  it  will  be  perceived  that 
the  exponents  of  e  form  an  ascending  series, 

0,   1,  2 .n—%  n— 1,  71. 

Then,  putting  V=  to  the  given  equation,  W=  to  the  sum 
of  the  coefficients  of  c°,  X=  to  the  sum^of  the  coefficients  of 

Y  Z 

e.  — =  to  the  sum  of  the  coefficients  of  e*. =  to  the  sum 

2  '2.3 

of  the  coefficients  of  c*,  &c.,  we  shall  have 

Y  =x"+6x'^'H-cx''-'4-<fe~-' ra^-fjx^+^x+tf^O; 

W=y'»+^-'+cy"-'+(/y"-^ ry^-\'sf-\-ty-\'U  j 

X  =ny"-'+(»— l)^y^+(7i— 2)cy'»-«  .3ry'+2*y+^- 

Y  =n(n— l)y^'+(n— l)(7i— 2)^*^  .6ry  +2*  ; 
Z  =n(»— l)(»-2)y'-'+(n-l)(7i-2) 

(n— 3)^-^* 6r,  &c. 

28 


32*6  ELEMENTS    OF    ALGEBRA.  [sECT.  XI. 

The  above  expressions  are  called  derived  polynomials,  and 
by  examining  them  we  may  readily  discover  the  manner  in 
which  they  are  derived.     Thus, 

1.  W  is  derived  from  V,  by  simply  changing  x  into  y. 

2.  X  is  derived  from  W,  by  multiplying  each  of  the  terms 
of  W  by  the  exponent  of  y  in  that  term,  and  diminishing  this 
exponent  by  1. 

The  above  law  will  be  found  useful  in  the  transformation 
of  the  higher  equations.  To  illustrate  its  application,  we 
will  subjoin  a  few  examples. 

EXAMPLES. 

1.  Transform  the  equation  a:"— ISaj^'+lTa?^— 9a:-f  7=0  into 
another  which  shall  want  the  second  term. 

Let  x=y-\-——y-\-^,  or  3+y.        / 
4< 

Substituting  this  for  x  in  the  given  equation, 

(3+yy-  12(3+yy+  17(3+yy-9(3+2/)  +  7=0.    - 

This  will  give  the  transformed  equation  of  the  4th  degree, 

and  of  the  form 

and  the  operation  will  be  reduced  to  finding  the  values  of 
these  coefficients. 

Now  it  follows,  from  the  preceding  law,  that 

W   '=      (3)^-12.  (3f +17.  (3f-9.(3y+7=-110; 

X     =4.(3)='-36.(3f+34..(3)'-9      .     .     =-123; 

Z     =6.(3f-36.(3y+17 =-37; 

At 

_A=,4.(3y-12 =         0. 

2.3  ^  ^ 

Therefore,  the  transformed  equation  becomes 

y4_37y2_i23y_i  10 =0. 

2.  Transform  the  equation  4a?^— 5a?^+7a7— 9=0  into  an- 
other, the  roots  of  which  shall  exceed  the  roots  of  the  given 
equation  by  unity. 

Let  y—x-\- 1,  then  x= — 1-f  y,  which  gives  the  transformed 
equation  / 


=  -h  4. 


SECT.  XI.]         GENERAL   THEORY    OP    EQUATIONS.  327 

But, 

W   =  4.(— 1)'—  5.(— 1)«4-7.(— 1)'— 9  .     =—25; 
X     =12.(— ly— 10.(— 1)'  +  7  ...        .     =  +  29; 

I     =12. (-1)-  5 =-17; 

^=4 

2.3 

Whence  the  transformed  equation  becomes 

4y'— 17y^-|-29y— 25  =  0. 
3.  Transform  the  equation  XT' — 10x*+7ar*+4x — 9=0  into 
another  which  shall  want  the  second  term. 

Let  x=y — — =y — 2,  or  — 2-|-y;   then  the  transformed 

0 

equation  becomes 

But, 

W       =        (2)'— 10  .  (2)*4-7 .  (2f +4 .  (2)'— 9=—  73 
X        =  5  .  (2)^—40  .  (2f  4-21 .  (2)^+4  .     .     =—152 

=  10.(2)^— 60.  (2)^+21.  (2)»  .     .     ,     =—118 


=  10.(2)'— 40.(2)' -h  7 =—33; 


Y 

2 
Z 

2.3 

_?.'_=  5.  (2)'—10 :    .    .     =        0. 

2.3.4  ^  ^ 

Hence,  the  transformed  polynomial  is 

y5_33y'_118y2— 152y— 73=0. 

4.  Transform  the  equation  3x'-f-15x^-|-25a? — 3=0  into  an 
equation  wanting  the  second  term. 

Divide  the  equation  by  3,  and  proceed  as  before. 

27 

5.  Transformtheequation3a:*— 13ar'+7x'— 8a:— 9=0  into 
an  equation  in  which  the  roots  shall  be  less  than  the  roots 
of  the  given  equation  by  |. 

An,.  3j,'-9y»-4y'-^y-H?=0. 


328  ELEMENTS    OF    ALGEBRA.  [sECT.  XI. 

Cor.  5.  If  a,  a\  a'  a'  ....  a"^  represent  the  n  roots  of 
the  general  equation  V=0,  or  a?"4-6a?"~'  +  ,  &c.,  ==0,  we  shall 
have,  by  Art.  453,  Cor.  2. 
a7''+Z?a?"-'+ca?"-2+,  ^^  ^  —[x—a)  {x—a)  {x—a') ....  {x—aT), 

Now  if  we  put  x=^y-\-e^  and  substitute  this  value  for  a?  in 
the  equation,  it  becomes 
(y^eY^rh{y^rey-'+,  &c.,  =(2/+e— a)  {y^e—a) . . .  {y^-e—oT), 

Or,  

(y+e)"+^>(2/+e)'*-^+,  (fee,  =(e+y— a)  (e-{-y—a) ....  (e-\-y—ar). 

The  first  member,  by  Cor.  4,  equals 

Xe+L^ e\ 

With  respect  to  the  second  member,  it  follows,  from  the 
preceding  theorem, 

1.  The 'part  involving  e  ,  or  the  last  term,  is  equal  to  the  prod- 
uct (y — a)  (y — a')  ....  (y — a"")  of  the  factors  of  the  proposed 
equation;  hence, 

W=(y—a)  (y—a) .  (y— «")• 

2.  The  coefficient  of  e  is  equal  to  the  sum  of  the  products  of 
these  n  factors,  taken  n — 1  and  n — 1,  or  equal  to  the  sum  of  all 
the  quotients  that  can  be  obtained  by  dividing  W  by  each  of  the 
n  factors  of  the  first  degree  in  the  given  equation  j  hence, 

V    w   ,    vv   ,    w  w 


y — a     y—a      y — a  y — a  ' 

3.  The  coefficient  of  e^  is  equal  to  the  sum  of  the  prod- 
ucts of  these  n  factors,  taken  n — 2  and  n — 2,  or  equal  to 
the  sum  of  the  quotients  that  can  be  obtained  bydividing  W 
by  each  of  the  factors  of  the  second  degree  >  hence, 
Y  W  W  W 


2     {y—a)  [y—a)     (y—a)  {y—a')  {y—a"'-')  {y-a"% 

CoR.  6.  If  two  or  more  of  the  roots  of  the  given  equation 
are  equal  to  each  other ;  that  is, 

«  =  «'=:«",  &c., 

the  derived  polynomial,  which  is  the  sum  of  the  products  n 
factors^  taken  n — 1  and  n — 1,  contains  a  factor  in  its  difTer- 


SECT.  XI.]    GENERAL  THEORY  OF  EQUATIONS.  329 

ent  parts,  which  is  two  or  more  times  a  factor  of  the  pro- 
posed equation. 

Hence,  if  the  equation  contain  equal  roots^  there  must  be  a 
common  divisor  between  the  first  member  of  the  proposed  equation 
and  its  first  derived  polynomial. 

460.  Problem. — Having  given  an  equation^  it  is  required  to 
discover  whether  it  has  equal  roots^  and  to  discover  the  method 
ef  determining  these  roots. 

Resume  the  general  equation  j  or,  since  the  polynomial 
W  differs  from  V  only  by  the  substitution  of  y  for  x, 
y^Jf-by''-^-\-cy'^^ ty-\-u-0. 

Then,  supposing  the  equation  to  contain  m  factors  equa] 
to  y  —  a\  &c.,  and  also  to  contain  the  simple  factors  y— />, 
y—qy  &;c.,  then  will 

W^Cy— a)'"  {y—a'y  {y—a'y  —  iy—p)  (y—q)  iy—r)  — 

Whence,  by  the  preceding  corollary, 

j^^mW     m'W     m'^W  WWW 

y-a     y—a      y—a" y—p     y—q     y—r 

Now  (y— a)"~^  (y— a')'"'~^&c.,  are  factors  common  to  all 
the  terms  of  the  above  polynomial  j  hence  their  product 

(y—a)'^-^x(y — a')*"'-^  x(y— a")'"'~* 

is  the  greatest  common  divisor  of  the  polynomials  W  and 
X;  or, 

D=:(y — a)'»'-^x(y — a')'"'-*  x  (y— a")'""~* > 

that  is,  the  greatest  common  divisor  is  composed  of  the 
product  of  those  factors  which  enter  two  or  more  times  in 
the  given  equation,  each  being  raised  to  a  power  less  by 
unity  than  in  the  given  equation. 

Hence,  to  discover  whether  an  equation  W=0  contains  any 
equal  roots^  form  X,  or  the  derived  polynomial  of  W ;  then  seek 
for  the  greatest  common  divisor  between  W  and  X  j  if  one  can- 
not be  obtained,  the  equation  has  no  equal  factors,  and,  const' 
quently,  no  equal  roots. 

461.  Again,  if  the  greatest  common  divisor  (D)  is  of  the 
first  degree,  or  of  the  form  y—a,  make  y—a  —  0,  whence 
y=a  ;  and  we  may  conclude  that  the  equation  has  two  roots 

Tt 


330  ELEMENTS    OF   ALGEBRA.  [sECT.  XI. 

equal  to  a  ;  if  it  is  of  the  form  [y — a)'*,  we  may  conclude  that 
the  equation  has  n-{-l  roots,  each  equal  to  a. 

If  the  greatest  common  measure  (D)  is  of  the  form 

we  must  find  the  two  values  of  y.  Let  a  and  a  represent 
those  values,  then  the  equation  will  have  two  roots  each 
equal  to  a,  and  two  each  equal  to  a. 

Hence,  the  equal  roots  of  an  equation  may  be  obtained  by 
finding  the  greatest  common  divisor  of  its  first  member  and  its 
derived  polynomial,  and  solving  the  equation  obtained  by  putting 
this  common  divisor  equal  to  0. 

EXAMPLE, 

Has  the  equation  aP — '7x'^-\-lQx — 12=0  equal  roots'?  if  so, 
how  many,  and  what  are  they  ] 

The  derived  polynomial  of  this  equation  is 

Performing  upon  this  and  the  first  member  of  the  given 
equation  the  operations  indicated  in  Art.  137  to  find  the 
greatest  common  divisor,  we  obtain 

x—% 

Then       -         -         -        a?— 2=0, 

And         -        '        -        X      *=2. 
Therefore,  we  conclude  the  equation  has  two  roots  equal 
to  2. 

Now,  since  the  equation  has  two  roots  equal  to  2,  it  must 
(Art.  460)  be  divisible  by 

(x—2y=x^—4>x-{-4>. 

Whence     -     a;^— 737^+ 16a7— 12=(a;— 2)'(a?-3)=0, 

And  -         -         -         -     'x —  3=0,  or  07=3, 
which  is  the  other  root  of  the  equation. 

462.  To  show  the  application  of  the  preceding  principles, 
we  will  subjoin  a  few  equations  with  equal  roots. 

EXAMPLES. 

1.  Reduce  the  equation 

2x'—12x^-\-19x''—6x+9z^0j 
which  has  eoual  roots. 


SECT.  XI.]         GENERAL   THEORY    OF    EQUATIONS.  331 

The  derived  polynomial  is 

8i»— 36aH-38a:— 6. 
Whence      -        -      D=«— 3=0, 
And     -        -        -        -    X      .  =  3. 
Therefore,  thfe  equation  has  two  roots  equal  to  3. 

Dividing  its  first  member  by  (x — 3y=x^ — 6a7+9,  we  obtain 

2x^+1=0,  OTX=±V'^h 
Hence,  the  four  roots  of  the  equation  are 

3,  3,  v^"^,  and  — v/^. 
2.  Reduce  the  equation 

x^— 2T*+ac'— 7x»+8x— 3 = 0, 
which  has  equal  roots. 

The  first  derived  polynomial  is 

5a:«_8ar'H-9x»— 14H-8. 
Whence       -        D=x'—2x+l,  or  (x—lf ; 
And  the  given  equation  has  three  roots  equal  to  1. 

Dividing  the  first  member  by  (x — 1)',  or  x' — 3x'+3x — 1, 
we  have 

a:*+x+3=0,  or  x=—i±is/—n. 
The  five  roots  are, 


1,  1,  1,  — i+iv/— 11,  and  _^-iv/— n.       " 
3.  Reduce  the  equation 

xH5x«-f6x^— 6x^— ISr*— 3a:»+8x+4=0, 
which  has  equal  roots. 

W=  x'4-5x«  +  6x'—  6x*— ISr*— 3a^+8x+4.,- 
X  =7x«+30x'+30x*— 24f'— 45x'— 6x+8; 
D  =  x*-\-  3jr'+    a^—  3x  —  2. 
Since  D  surpasses  the  second  degree,  we  must  apply  to  it 
the  same  process  we  have  to  W. 
Its  first  derived  polynomial  is 

4r'+9x*+2x— 3, 
And  the  greatest  common  divisor  ;  or, 
D'=a:+1. 
Hence,  D  has  two  equal  roots  equal  to  — 1 ;  and,  dividing 
it  by 

(x+l)»,  or  x*+2a;+l, 
we  have        •        x'-|-x — 2=0  5  or  x=l,  or  — 2. 


33!^  ELEMENTS    OF    ALGEBRA.  [sECT.  XII. 

Therefore, 

D,  or  x'+3x'+x''—3x—2={x+lf{x—l)  (.r+2), 
And  -        -         -  W  =  {x+lf{x—lY(x+2f. 

The  roots  of  the  equation,  then,  are 

1,   1,  —1,  —1,  —1,  —2,  and  —2. 
4.  Required  the  equal  roots  of  the  equation 
x'—Sx'-]-26x'—4>ox'-\'4^bx'—21x'—10x'^20x—^z^0. 

Ans.  1  and  2. 


SECTION  XIL 

RESOLUTION  OF  THE  HIGHER  EaUATIONS. 

Resolution  of  the  Cubic  Equations  ly  Cardan's  Rule. — Young's 
Method.  —  Des  Cartes'*  Method  of  Resolving  Biquadratic 
Equations. — JVewton^s  Method  of  ^Approximation. — Resolu- 
tion of  Higher  Equations  by  Trial  and  Error, 


463.  We  will  now  proceed  to  investigate  the  methods  by 
which  affected  equations  of  the  third  degree  may  be  solved. 
Equations  of  this  nature  may  all  be  exhibited  under  the  ttiree 
following  forms,  in  which  p,  p\  and  g  may  be  either  +  or  —  : 

(1.)     x^-\-px  =q; 

(2.)     x'+px'         =.q; 
(3.)     x^-\-p'x'^-\-px  =  q. 
JSTote. — The  known  quantities  p,  p\  and  q  are  here  used  iii 
their  most  general  sense,  and  may  be  entire  or  fractional, 
positive  or  negative  quantities. 

First  Form. 

464.  In  order  to  deduce  a  general  formula  for  the  reduc- 
tion of  cubic  equations  of  the  first  form,  let  us  take 

x^-\-px  =  q. 
Let         -         y-\-z—x^  and  Syz= — p; 
Then      -        -    ^=(y+zY=f+3fz+Syz'+z'i 


SECT.XIl]    AFFECTED  EQUATIONS  OF  THIRD  DEGREE.  333 

Resolving  into  factors        -      ar'=y'-|-3y2(y-f  «)+2'; 
Substituting  X  for  y+2       -       a?*=y'4-3y2X+2^} 

Substituting  this  value  of  x^^   y^^z'-}-3yzx-\-px=q  ; 

in  the  1st  equation  •  S 

JResolving  into  factors        -      y*-f  2*H-(3y2-f^)x=g, 
Or     -         -         -     ^    -         r       y^-{-z^+{—p-\-p)x=q; 
Whence     -         -     *   -         -       y^+z*  =q* 

To  determine  the  values  of  y  and  2,  we  have  the  two 
equations, 

y»H-2»=y;  (L) 

3yz=^p;  (2.) 

Dividing  the  2d  3       -        -        yz  =—lp ;  (3.) 

Cubing       .         .         -         -       f2^=-^\jp»;  (4.) 

Squaring  the  lst7       yS^2fz^+z'=f  j  (5.) 

equation  -  5 

Multiplying  the  4.th  by  4    -     4^2*=: — ^\p^;  (6.) 

Subtracting  the  6th  )  ^.o^a^+^e^  2^  ,    3  /^  x 

from  the  5th  -      i  ^       ^  ^  ___ 

Extracting  the  square  root,    y'' — z^=±  Vq^+-sSp\    (^O 

Or y^— r»^±2v/ig^+^S/.(9.) 

Adding  the  9th  to  the  1st  -        ^f=q±^\^iq'-\-^\p^y 

3/ ===r 

Dividing  and  evolving        -  y=\/ i9^  '^i9^'^iSP^i 

Subtracting:  the  9th  from  the  }      ,  ^    — 3 r- 

1st  !         .         .         .     1  ^2^=q^^2V}q'+^\p'; 

Dividing  and  evolving        -  ^=\/h9^  >/W'^j\P^9 

3/ , 

Consequently,  we  take        -  y=\/i?+ v^4?^+lV?^ 

And «=\/k— v/T?+SP- 

Adding  the  last  two  equations,  and  observing  that  y-[-2z=x, 

^=\/iq+  ^W+i^'+\/k-^iq'-^i\p*    (A.) 

Or,  since  2= — 2E  and  x=y+2, 

y 


Y/ij+v/ie^+^P'-j^ 


\^k+^^W-^i\p'-  (A^) 


834  irLEMENTS    OF   ALGEBRA.  [sECT.  XII. 

Again,  taking  the  equation  x^—px—q^  and  letting  y-{-z—x, 
and  3yz=+j9,  and  proceeding  as  before,  we  shall  obtain* 

x=^hq+  Viq'-j\f-\-\/-hq-Vh'-^\p\      (B.) 

Or,  since  z—'^,  and  x=:y-\'Z, 


^=\/h+Vif-^\f+- 


ip 


By  the  above  formulas  we  may  obtain  the  exact  or  ap- 
proximate roots  of  cubic  equations  of  the  first  form.f 

EXABIPLES. 

1.  Find  the  value  of  a:  in  the  equation  x^+6x—2. 
Substituting  6  for  jo,  and  2  for  g,  in  formula  A,  we  have 

Whence 

a?z.r  ^/4+^^=  1,587401— 1,259921=,32748  +  .     jlns. 

2.  Find  the  value  of  x  in  the  equation  x^ — 2a:— — 4. 
By  formula  B,  we  have 

x=  \  Ai+  v/-L«.— 8-4-  \/z^— n/JJ.— _8_  : 

V       2  4  27  \       2  4  27 

=  y_2+ioy  3+y^-2-J-0  V3  J 

=  ^— 2+l',9245  +  e^=^2^1";9245  j 
=z^^=;o755—e/3,9245z=— ,41226-  1,5773} 
=—1,9999  +  ,  or  —2.     ^ns, 

*  In  formirias  B  and  B',  it  is  evident  that,  if-^^p^'^^q^,ihe  equation  can- 
not be  reduced,  since  it  involves  the  extraction  of  the  square  root  of  a  neg- 
ative quantity;  hence,  the  value  of  x  can  only  be  obtained  by  imaginary 
quantities,  and  the  conditions  of  the  question  are  incompatible  with  each 
other. 

t  These  formulas  are  substantially  what  is  known  under  the  cognomen 
"  Garden's  Rule  for  Cubic  Equations.'?  The  invention  of  the  rule,  however, 
is  due  to  Nicholas  Tartalea  and  to  Scipio  Ferreus,  who  found  it  independ- 
ently of  each  other;  but  Garden  first  published  it  to  the  world. — See  Ed. 
Encycl.,  Art.  Alg. 


SECT.  XII.]    AFFECTED  EQUATIONS  OF  THIRD  DEGREE.  335 

3.  Find  the  value  of  xin^he  equation  x* — 6x=12. 

By  formula  B',  we  have 

3  / 2 

a?=\/ll-f  v/JJ.'+z±«-h —  =3,1392.    Ans, 

V     a      '  4  2  7 

4.  Find  the  value  of  x  in  the  equation  o? — 15x=4*. 

Ans,  x=:4. 

5.  Find  the  value  of  a:  in  the  equation  x'-|-9x=584'. 

Ans.  x=8. 
Second  Form, 

465.  If  the  second  term  be  made  to  disappear  from  a  cubic 
equation  of  this  form,  there  will  result  a  cubic  equation  of 
the  first  form  (Art.  459,  Cor.  1). 

Hence,  to  reduce  equations  of  the  second  form,  we  have 
the  following  general 

EULE. 

Transform  the  given  equation  into  one  of  the  first  form^  and 
then  reduce  as  before. 

EXAMPLES. 

1.  Find  the  value  of  x  in  the  equation  x'+3x*=54. 

Let x=z — l  =  z — Ij 

Then af'=z'—32^-{-3z—l, 

And 3x»=    +.32;'— 62+3; 

Adding  the  two  equations,  x*+3x'=2r' — 3z  +2; 
Hence   -        -        -  2'— 32+2=54 ; 

Transposing  -        -        -       2* — 32=52. 
Applying  formula  B,  we  have 

V  2^     4      n^y   2        4  ^T?* 

3/ =izzr    3/ zzzzzz 

=  \/26+\/2_l±4_-?_I+\/26  — v'iUL* u  ; 

V  4  aT  y  4*7 

=  y/26+v/676^+  y/26— v/676— 1  j 

=  ^26+25,980761921+^26-25,980761921  j 


336  ELEMENTS    OF    ALGEBRA.  [sECT.  XII. 


3. 


1  8^ _3^ 

2  7 


=:^51,980761921+>2/,019238079  ; 
=:3,732+,267=i  3,999. 
Hence  jt zn 3,999 -l=z 2,999+,  or  3.     Ans. 

2.  Find  the  value  of  x  in  the  equation  x^ — 3a?^=:  16 

Let        -        -         x—z-\-%—z-\-\ ; 

Then     -        ^—fz^  IG+^V  X  ^\  or  ^  — 32^=  18. 

Applying  formula  B'  to  this  last  equation, 

3/             ZZIZZI.               3 
Z—\/uLa_s/11} 1L\ 2 

V       2  4 

Hence  a?=2:+ 1  =  34- 1=4.     •^'i*- 

3.  Find  the  value  of  a?  in  the  equation,ir^+6a;^=1600. 

Ans.  10. 

4.  Find  the  value  of  x  in  the  equation  x^ — ,3a7^=^-,004. 

Ans.  \2. 

Third  Form. 
x^-\-p'x^-\-px=zq. 

466.  Making  the  second  term  disappear,  we  shall  have,  as 
before,  an  equation  of  the  first  form  ;  hence,  the  method  of 
reducing  an  equation  of  the  second  form  will  be  the  same  as 
that  for.the  second  form. 

EXAMPLES. 

1.  Find  the  value  of  x  in  the  equation  x^ — 6a?^+ 18a?  ==22. 
Let  x=z-{-^—z-\-2f  then  we  shall  have  (Art.  459) 

2^-\-6z=2. 
Applying  formula  A,  we  shall  find  z=  v^4 — v^2. 
Whence 
a?=z+2=^4— ^2+2=1,5874— 1,2599+2=2,3274.    Ans. 

2.  Find  the  value  of  x  in  the  equation  x^-\-Sx'^ — 4a? =32. 

Ans.  a? =2. 

3.  Find  the  value  of  x  in  the  eq'^iation  a^ — 10a?^+10a?=100. 

Ans.  x=10. 


SECT.  XII. ]    RESOLUTION    OP    HIGHER    EQUATIONS.  337 

young's  method  OF  RESOLVING  CUBIC  EQUATIONS. 

467.  Every  cubic  equation  may  be  transformed  so  as  to 
appear  under  the  form 

r'-f6a^+ca:=N.  (A.) 

468.  Now,  suppose  that  two  consecutive  numbers  in  either 
of  the  series 

1,  2,  3,  &c.,  or  10,  20,  30,  frc,  or  0 .  1,  0 . 2,  0 . 3,  &c., 
are  found  such,  that,  substituting  the  first  for  x  in  the  above 
equation,  the  result  shall  be  less  than  N,  and,  by  substituting 
the  second,  the  result  shall  be  greater  than  N  ;  then  the  first 
of  these  numbers  will  be  the  first  figure  of  one  of  the  roots 
of  the  equation.  Let  this  figure  be  represented  by  r,  and 
the  other  succeeding  figures  of  the  same  root  by  $,  t, «,  &c. ; 
then,  substituting  for  x  the  first  figure  (r)  of  its  root  in  the 
equation  (A),  we  shall  have 

r»+^,7-»-fcr=N;  *(B.) 

Whence         -        -        -     r=—^--,  (C.) 

469.  Let  the  remaining  figures  of  the  root  equal  y,  then 
x=r-\-y:  substituting  this  value  for  x  in  the  first  equation 

(A),  we  have 

cy-f-cr  =cx 

bf^2bry-^br': 
f+Srf-^-Sr'y 

Adding, y*+(3r+W+(3'^+2ir-|-c)y-h(r'+ir»+cr)=N.  (D.) 
But,  if        -        6'  =3r  +A,  (1.) 

c'  =3r»+2ir+c,       (2.) 

N'=N-r»-ir'-cr,  (3.) 
the  above  equation  becomes 

y34.fty+cy=N'.  (E.) 

470.  This  equation  is  in  all  respects  similar  to  the  first 
(A) ;  and,  since  s  is  the  first  figure  of  the  root  y  of  this 
equation,  substituting  as  before. 

Whence        -        -  s=-^-^ — ;•  (^0 

29  U  ^ 


f-^cr  =cx   \ 
r4-Jr»=ii»  )  =:N. 
+  r'=x'    ) 


338  ELEMENTS    OF   ALGEBRA.  [sECT.  XII. 

Supposing  the  value  of  s  found,  and  putting  t,  u^  &c  ,  equal 
to  2r,  or  y—z-rs,  we  have 

c'z-\-c  s=c'y  ; 

Adding,  !_' 

z'+{3s-\-¥)z'+(3s'-\-2Vs-]-c')z+(s'  +  b's'-\-c's)^'N\    (G.) 
But,  if        -         b''=Ss+b\  (4.) 

c''  =^Ss'-\-Ws+c\        (5.) 
W=W—s'—b's'—c's,{6.) 
the  above  equation  becomes 

z'-\-b''z'  +  c''z=W\  (H.) 

an  equation  which  is  in  all  respects  similar  to  the  first. 
Hence  we  may  proceed  in  the  same  way  to  find  the  first 
figure  f,  in  the  root  z,  and  so  on  till  we  have  found  all  the 
figures  in  the  root  x  of  the  proposed  equation. 

471.  Now,  by  observing  the  formation  of  the  coefficients 
b\  c'  in  the  equation  marked  (F),  and  recollecting  that  r, 
being  the  first  figure  of  the  root,  must  be  greater  than  s,  it 
will  appear  obvious  that  c'  must  form  a  part  of  the  divisor 
s^-\-b's+c\  and  if  r  be  already  known,  the  value  of  c'  will 
become  known  (2),  which  may,  therefore,  be  used  as  a  trial 
divisor  for  finding  s  ;  the  same  may  be  observed  of  the  next 
and  the  succeeding  divisors  j  but  these  trial  divisors,  c'',  c"', 
&c.,  will  continually  approach  nearer  the  true  divisors. 

472.  Now,  if  the  first  figure  of  the  root  r  be  found  by 
trial,  and  r-\-b  be  multiplied  by  it,  and  the  product  added  to 
c,  the  sum  will  be  the  first  divisor  ;  thus, 

r{r+b)=r'-\-br 

7'^-\-b7'-\-c=:  1st  divisor.  (7.) 

u  N  N 

Hence         -  r=  ——       ,  ,,  .    - 

r-1roj'-\-c     r{r-\-b)-\-c 

If  under  these  two  expressions  we  write  r',  and  add  up 
the  three,  we  shall  obtain  c';  thus, 


SECT.  XII.]     RESOLUTION    OF   HIGHER   EQUATIONS.  339 

r* 


3r«+2Z»r+c=c'.  (8.) 

Having  obtained  c',  we  have  a  trial  divisor  of  N'  that  will 
enable  us  to  determine  more  easily  the  next  figure  s  of  the 
root. 

When  s  is  found,  the  second  divisor  may  be  computed ; 
thus, 

s*-^-b's-\-c'=  2d  divisor. 

Hence    -        -        s= ^ = ^1 

s'-{-b's-\-c'     s{s-\-3r+b)-{-3r'-\'2br^t 

By  a  similar  process  we  shall  obtain 

/'+*"/+c"=  3d  divisor. 


Hence,       /= 
Also, 
Hence,  tt= 


Also,  tt'+6'"tt+c'"=  4th  divisor. 


&c.,  Scc\,  &c. 

The  above  formulas  may  be  readily  applied  to  the  reduc- 
tion of  cubic  equations.  By  a  careful  inspection  of  them,  we 
may  obtain  the  following  general 

RULE. 

1.  Put  down  c,  the  coefficient  of  a?,  and  a  little  to  the  right 
place  the  absolute  number^  which  is  to  be  considered  as  a  divi' 
dend,  the  figures  of  the  root  forming  the  quotient. 

2.  Place  the  first  figure  of  the  root^  found  by  trials  in  the  quo- 
tient^ above  which  u^ite  the  coefficient  of  a?-,  observing  that  its 
unites  place  he  over  the  unit's  place  of  the  quotient. 

3.  Multiply  the  value  of  the  quotient  figure,  taking  in  those 


340  ELEMENTS    OF    ALGEBRA.  [sECT.  XII. 

above  by  that  value  ;  add  the  product  to  c,  and  the  sum  is  the 
first  divisor, 

4.  Write  the  square  of  the  quotient  figure  just  found  under 
the  first  divisor^  add  it  to  the  two  sums  immediately  above^  and 
the  result  will  be  the  trial  divisor  for  finding  the  next  figure  of 
the  root. 

5.  Find  now  the  next  figure  of  the  root^  and  to  its  value  (in- 
cluding those  above  it)  prefix  three  times  the  preceding^  taking 
the  value  of  the  figure  above  it ;  multiply  the  result  by  the  last 
found  figure  ;  add  the  product  to  the  trial  divisor ^  and  we  shall 
have  the  true  divisor  ;  and  in  the  same  manner  are  the  succeed- 
ing divisors  to  be  obtained. 

EXAMPLES 

1.  Reduce  the  equation  a:^-f8a:^-f  6a7=75  .9. 


SECT.  XII.]     RESOLUTION   OP   HIGHER   EQUATIONS. 


3^1 


§  H- 

Id 

CI 


r-t      00 

II  i 


3 


+ 


3 
CO 

1        II 

•E  '« 

CO    v** 


kt 


5  «+  ^ 

IS  "»     Z 

II 


11  II 

II      II 

II  II 

CO 

00 

<N  t- 

GO 

^  c^ 

CO 

CO  00 

^^ 

CO    C5 

CO     ^ 

^5 

Oi    00 

lO    ^ 

Oi   00 

kO   c< 

CO    00 

00    (?< 

1-^   1-^ 

o  o 

(M   <N 

CO 

5 


cj 


3 

cr 

V 


o     o       ^ 

«       o      4> 

:>  ^  s 


CO    O       CO    ^ 


lO 

>r> 

c< 

c* 

3 

5S 

d  CO 

in 

05     CO 

kO 

SP 

CO    CO 

00   o 

00   o 

oo  t- 

CO 

t- 

t-  .^ 

CO  CO 

Oi    O 

(N    O 

00 

o  i« 

Id 

r-^ 

r^     O 

c<  o 

CM 

»o 

\Ci 

CO 

CO 

CO 

CO 

II  II    If  II    II  il 


II    II  II 


00 

+ 

c<      '     • 

II        l^ 


C<    00 

+  4- 


II  z 


Cd 

+     + 


C         •? 


•a 

C4 


CO     ^       CO 

[I         4- 


G\    00 

+    + 

il    » 


^  o 

•*  oo 

II  > 

s  ^ 


CO 


CO 


342 


ELEMENTS    OF    ALGEBRA. 


[sect.  XII. 


J^ote. — By  inspecting  the  preceding  example,  we  shall  ob- 
serve that  if,  after  obtaining  three  places  of  decimals  in  the 
right-hand  column,  we  had  continued  to  reject  the  remain- 
ing decimals,  we  should  have  had  the  root  equally  correct 
to  three  places  of  decimals.  Now,  in  order  that  the  num- 
ber of  decimals  in  the  last  column  may  not  exceed  three,  it 
is  obvious  that  the  divisor  corresponding  to  the  first  deci- 
mal in  the  root  must  contain  but  two  decimals,  that  corre- 
sponding to  the  next  decimals  of  the  root  but  one,  and  that 
for  every  succeeding  decimal  in  the  root  the  right-hand  di- 
git of  the  corresponding  divisor  must  be  cut  off.  It  should, 
however,  be  observed,  that  whatever  would  have  been  car- 
ried had  the  complete  multiplication  been  performed,  is  still 
to  be  carried  for  the  mcrease  of  the  next  figure  5  and,  indeed, 
if  the  figure  cut  off  exceed  5,  one  is  to  be  carried  to  the 
next  figure. 

Hence  the  work  of  the  above  example  may  be  rendered 
more  concise,  and  will  stand  as  follows,  the  figures  cut  off 
being  placed  a  little  to  the  right : 

6  75.9  12.4257+ 

20 


52 


26 
4 

"50 
5.76 


55.76 
.16 

./ 

61.68 

.3a 

44 

61.9 

844 
004 

62.3 
1 

892 
76325 

23.9 
22 .  304 

1.596 
1.240 

688 

.356 
.312 

312 

827625 

.044 

484375 

62.4 


SECT.  XII.]     RESOLUTION    OF    HIGHER    EQUATIONS.  343 

2.  Reduce  the  equation  a:'-f.x'=500. 
This  equation  is  the  same  as  x'-f  x'-}-0x=500 ;   hence 
i=l,  c=0,  and  N=500. 

The  first  figure  of  the  root  is  7. 


0 
56 

56 
49 

500 
392 

108 
104. 

|7. 61727975,  &c.,=x. 
736 

161 
13.56 

3 
1, 

.264 
.887181 

174.56 
36 

1. 
1, 

.376819 
. 323862 

188.48 
.2381 

52957 
37859 

188.7181 

1 

15098 
13251 

188 . 9563 
1669 

1847 
1704 

189 .  123|2 

189 .  290 
5 

143 
133 

10 

.  iiQio    loioi;;, 

9 

3.  Reduce  the  equation  x* — 17x'-|-54a:=350. 

An8.  1=14  .  954,  &c* 

4.  Reduce  the  equation  x'-f2x'+3x=  13089030. 

An8,  a?=235. 

5.  Reduce  the  equation  j^-|-2x^— 23x=70. 

Ans.  a:=5  .  1345,  &c. 

6.  Reduce  the  equation  a:* — 2x=5. 

^iM.a;=2»  09455 14815423265917,  &c. 


344  ELEMENTS    OF   ALGEBRA.  [sECT.  XII. 

DES  cartes'  method  OF  RESOLVING  EQUATIONS  OF  THE  FOURTH 
DEGREE. 

473.  Every  equation  of  the  fourth  degree  may  be  reduced 
to  the  form 

x*-\-bx^-\-cx''-\-dx+e=zO.  (A.) 

This  equation  may  also  be  transformed  into  another  which 
shall  want  the  second  term  (Art.  459)  j  thus, 

x'-]-c'x^-{-d'x+e'=zO.  (B.) 

474.  Now  if  we  can  arrive  to  a  solution  of  the  equation  in 
this  form,  in  which  the  roots  sustain  a  given  relation  to  the 
original  equation  (A),  the  complete  solution  of  that  equation 
may  be  effected. 

Now,  suppose  B  to  be  formed  by  the  product  of 
x'+px+q^O,  (1.) 

x^-i-rx-{-s=0,         (2.) 
two  equations  in  which  p,  q,  r,  and  s  are  unknown  quanti- 
ties, and  we  shall  obtain  by  the  actual  multiplication  of  the 
factors  (1)  (2),  and,  taking  the  sum  of  the  coefficients  of  the 
equal  powers  of  a:, 

co'+{p+r)af'-^{s-hq+pr)x'+{ps-[-qr)x-{-qsz=zO.     (C.) 

Whence     -    p+r         =0,  or  r= — p;     (3.) 

s+q-{-pr=c';  (4.) 

ps+qr       =d'}  (5.) 

qs  =e'.  (6.) 

Or,  substituting  — p  for  r  in  (4)  and  (5),  and  transposing, 

they  will  become 

S+q^c'+p'-,  (7.) 

S-q=t  (8.) 

P 
And,  by  subtracting  the  square  of  (8)  from  the  square  of 
(7),  we  have 

c'2+2c>2_j_^4_»  ^4^5^  or  4e'., 
P 
Or,  clearing  of  fractions,  and  arranging  the  terms  with 
reference  to  the  highest  power  of  ^,  we  have 

/+2cy+(c'2_4eV-(/'2:=0.  (D.) 


SECT.  XH.]     RESOLUTION    OP    HIGHER    EQUATIONS.  346 

li  p*z=z,  this  equation  will  become 

,  z^+ac'z^-f-Cc''— 4«>— ^"=0.  (E.) 

Now,  if  we  add  and  subtract  equations  (7)  and  (8),  and 
divide  the  result  by  2,  we  shall  have 

,=  ic'+ii^+±;  (9.) 

9=ic'+iF'-~         (10-) 

4-75.  From  these  two  formulas  (9)  and  ( 10),  p  being  known 
from  equation  (E),  s  and  q  can  be  obtained. 

Hence,  substituting  —p  for  r  in  equation  (2),  and  reducing 
the  two  equations  (1)  and  (2),  we  shall  have 
X=-i,p±Vj^^q;  (11.) 

X=  +  ip±Vip'-s.  (12.) 

These  equations  (11)  and  (12)  give  the  four  roots  of  the 
biquadratic  equation  (B). 

Cor.  1.  The  cubic  equation  (E)  gives  three  roots;  but  the 
same  values  of  x  will  be  obtained,  whichever  of  the  roots  be 
used. 

Cor.  2.  If  the  roots  of  the  cubic  equation  (E)  are  all  real, 
the  roots  of  the  biquadratic  equation  (B)  will  be  real  also. 

If  only  one  root  of  the  cubic  equation  (E)  be  real,  then  the 
proposed  biquadratic  (B)  will  have  two  real  and  two  imaginary 
roots. 

476.  The  above  formulas  may  be  readily  applied  to  the 
reduction  of  equations  of  the  fourth  degree. 

EXAMPLES. 

1.  Reduce  the  equation  x*— 3a:*H-6x+8=0. 

Comparing  this  equation  with  formula  (B),  we  shall  have 
c'= — 3,  (/'=r6,  and  e'=:S;  and  substituting  these  values  (oi 
c',  d\  and  e'  in  formula  (E),  it  becomes 
2^-62*-- 23z-36=0. 

Reducing  this  equation,  2=9  j  hence  p=y/z=V^=~-:t3» 

Substituting  -|-3  for;?  in  formulas  (9)  and  (10),  we  have 

«=ic'H-i;>«+^=~^+^4-f=4; 


346  ELEMENTS    OF   ALGEBRA.  [sECT.  XII. 

Substituting  these  3,  4,  and  2  for  7?,  s,  and  q  in  formulas 
(11)  and  (12),  we  have 

X::=z—ip±  Vif^9-=^-^±  v/f=:2  =  -|±i=:— 1,   Or  —2  ,' 

Hence  the  four  roots  of  the  biquadratic  equation  are  — 1, 

-%  l+lV^T,  and  i-v/^=7. 

2.  Reduce  the  equation  x*—4,x^—8x-{-32—0. 

Ans.  4,  2,  —  1  +  v^^^,  and  —\—^~—3, 

3.  Reduce  the  equation  o:^— 9a:^+30a;2— 46a?+24  =  0. 

*>  ^7i5.  1,  4,  2+  y^^,  and  2— v/^^. 

4.  Reduce  the  equation  a:' + 1 6a;' + 9 9a;^4- 228a; +144=0. 

Arts.  —1,  —3,  — 6  +  2v^^^,  and  —  6— 2v/"^. 


477.  This  is  an  expeditious  method  of  finding  the  approx- 
imate root  of  an  equation,  when  its  near  root  is  given  or  has 
been  ascertained  by  trial,  and  is  equally  applicable,  whatever 
be  the  degree  of  the  equation. 

478.  Let  us  resume  the  general  equation 
a.n_^j^n-i_^^^«-2_|_ *a;'4-^a;4-^~.0.       (A.) 

Then  let  a  represent  the  near  root  of  the  equation  which 
is  known,  and  z  represent  the  part  to  be  added  to  make  the 
root  complete  \  then 

a?=a  +  S.     (1.) 
Substituting  this  value  for  x  in  the  first  equation  (A),  we 
have 

(«+2:)"+Z>(a4-z)"-'  +  c(a+2r)"-2+ 5(a+2f)2+jf(a+2r)  + 

M  =  0.  (B.) 

Then,  transforming,  as  in  Art.  459, 

W+X^+I^^+^^ z-=.^,  (C.) 

479.  Now,  since  r,  by  hypothesis,  is  a  proper  fraction,  the 
terms  that  involve  z^^  z^^  &;c.,  being  less  than  2,  may  be  re- 


SECT.  XII.]     RESOLUTION    OF    HIGHER   EQUATIONS.  347 

jected  from  the  equation  without  departing  far  from  rigid 
exactness.     The  equation  (C)  will  then  become 
W  +  X2=0.  (2.) 

W 
Whence        -        -       ^=~^'      ^^'^ 

But,  comparing  with  the  transformations  in  Art.  459,  we 
shall  find 

W=  or    +bar-'  -{-car-* «a'-f/a+w;  (4.) 

X  =na'^'  +  {n—l)bar-^+(n—2)ca'*~^.  .  .  2sa  +e.  (5.) 

Substituting  these  values  for  W  and  X  in  equation  (3),  we 

have 

aT-^-ba^'^  +  car-* sa^-^ta-\-u  ^jv  v 

^~     na''-'-{-{n-i)ba''-'-\-{n-2)car^ 2sa+t'  ^    '' 

The  numeral  value  of  this  expression  should  be  calculated 
to  within  one  or  two  places  of  decimals,  and  added  to  the 
root  (a)  found  by  trial.  Let  the  resulting  approximate  root 
be  represented  by  a',  then  a'=a-{-z  ;  and  if  z'  represent  the 
part  still  to  be  added,  we  shall  have 

a"'+ba'^'-\-ca'^^ sa'^+ta'-{-u  ^ 

~     na'^'+(»— l)6a"*^H  (n—2)ca"^ 2sa'+i  ^ 

Letting  a"  represent  the  third  approximate  root,  we  shall 

have 

a"=a'+z', 

480.  Proceeding  in  this  manner,  the  approximation  may 
be  carried  to  any  assigned  degree  of  exactness. 

EXA3IPLES. 

1.  Reduce  the  equation  r'-f-2x^—8x=24. 

By  making  trial  of  1,  2,  3,  and  4,  we  shall  find  that  the 
root  of  the  equation  is  between  3  and  4,  and  very  nearly 
equal  to  3. 

Then  n  =  3,  a=3,  b  or  5=2,  c  or  ^=—8,  u= — 24,  and  Xz= 
a+z. 

By  substituting  these  values  in  formula  (D),  we  have 

z=— ?!±^  •  ^'~:?-  ^~"--=: A  =  0.09  j  hencex=3.09,nearly. 
3.3^+4.3—8         ^r  > 

Again,  if  3 .  09  be  substituted  for  a'  in  formula  (D'),  we 

shall  have 


348  ELEMENTS    OF    ALGEBRA.  [sECT.  XII. 

3.(3.09/+4.(3.09)— 8  ' 

a?=3.  09364. 

2.  Reduce  the  equation  x^-\-x^-{-x  =  90. 

Here  7^=3,  b  or  s=zl,  c  or  ^=1,  w= — 90,  and  a  will  be 
found  =4< ;  hence  x=4f-\-Zj  and  we  shall  have 

^=— i!±iji±ri?=:/^  =  0.  1 J  hence  a:=4. 1,  nearly. 

3.4^  +  2.4+1     ''  '  »  ^ 

Again, 
^.^_(4.ir+(4.ir+(4.1)-9_0^Q   QQ,33      hence  .=.4. 
3.(4.1)+2.(4.1)+1 
10283. 

3.  Reduce  the  equation  a;*— 38a:'+210j:=^+538a:+289=:0. 

^7X5.07=30.535653. 

4.  Reduce  the  equation  x'-\-6x^—10x'—n2x''—201x-\-110 
=0.  ^ns.  07=4.46410161. 

RESOLUTION  OF  HIGHER  EaUATIONS  BY  TRIAL  AND 
ERROR. 

481.  The  roots  of  cubic  equations  may  also  be  found  to 
a  sufficient  degree  of  exactness  by  successive  approxima- 
tions. From  the  laws  of  the  coefficients,  as  stated  in  Art. 
455,  it  is  evident  that  the  roots  must  be  such  that,  when 
their  signs  are  changed,  their  product  shall  be  equal  to  the 
last  term  of  the  equation,  and  their  sum  equal  to  the  coeffi- 
cient of  the  second  term.  By  considering  this  law,  some 
estimate  may  be  formed  of  the  values  of  the  roots,  and  a 
trial  may  then  be  made,  by  substituting  in  the  place  of  the 
unknown  letter  its  supposed  value.  If  this  proves  too  small 
or  too  great,  it  may  be  increased  or  diminished,  and  the  tri- 
als repeated  till  one  is  found  which  will  nearly  satisfy  the 
conditions  of  the  equations. 

482.  Now,  since  the  errors  in  the  results  will  be  very 
nearly  proportioned  to  the  errors  in  the  assumed  numbers, 
after  we  have  assumed  two  approximate  values,  and  calcu- 
lated the  errors  which  result  from  them,  we  may  obtain  a 
more  exact  correction  of  the  root  by  the  following  propor- 
tion : 


SECT.  XII.]     RESOLUTION    OP    HIGHER   EQUATIONS.  349 

The  difference  of  the  errors :  to  the  difference  of  the  assumed 
numbers  : :  the  least  error  :  to  the  correction  required, 

E'er,  letting  N  and  n=  the  assumed  numbers,  S  and  s= 
the  errors  of  these  numbers,  and  R  and  r=  the  errors  of  the 
results,  we  shall  have 

R  :  r  : :  S  :  5  very  nearly. 

Hence,  by  Art.  369,  R-r  :  S-5  i:r:s, 

483.  If  the  value  which  is  first  found  is  not  sufficiently 
correct,  this  may  be  taken  as  one  of  the  numbers  for  a  sec- 
ond trial  J  and  the  process  may  be  repeated  till  the  error  is 
diminished  as  much  as  is  required. 

There  will  generally  be  an  advantage  in  assuming  two 
numbers  whose  difference  is .  1,  or  .01,  or  .001,  &c. 

EXAMPLES. 

1.  Reduce  the  equation  ar'—8x^-f  1707—10=0. 

Here  the  signs  are  alternately  positive  and  negative, 
therefore  (Art.  455)  the  roots  must  all  be  positive  ;  their 
product  =10,  and  their  sum  =8. 

Suppose  J7=5  .  1  or  5  .  2  ;  then, 

By  1st  supposition, 

:! 


(5 .  l)'-8 .  (5  . 1)'+ 17 .  (5  . 1)- 10  =  1 .  271 .  , 

^        '  ^        '  ^        '  ^  errors. 


By  2d,  (5  . 2)='-8 .  (5  . 2)*+ 17 .  (5  .  2)- 10  =2  .  688 


Difference  of  errors   -        -        -        -      1.417 

Then      -        -       1  .  4  :  0  .  1 : :  1 .  27  :  0  .  09. 

Hence     -         -       a:=5  .  1-0  .  09=5  .  01,  nearly. 

To  correct  this  farther,  suppose  a?=5  .  01,  or  5  .  02;  then, 

By  1st  supposition, 

(5.01)»-8.(5.01)»+17.(5.01)-10=;0.121> 
By2d,(5.02)'-8.(5.02y+n.(5.02)-10=0.246  S 

Difference  of  errors   -        -        -        -        0.125 
Then      -        -       0  .  125  :  0  .01 : :  0  .  121 :  0  .  01. 
Hence    -        -      a:=5  .  01— 0  .  01=5. 
This  value  of  x  satisfies  the  conditions  of  the  equation;  for, 
5'-8x  5^-1- 17x5—10=0. 
Therefore,  one  of  the  roots  of  the  equation  is  5. 

30 


350  ELEMENTS    OF    ALGEBRA.  [sECT.  XII. 

To  find  the  other  two  roots,  let  the  first  member  be  divi- 
ded by  X — 5  (Art.  453),  and  the  quotient  put  equal  to  0. 

a?3_8x2+17a?— 10  |a?— 5 

x^ — 5a?^  x^ — 'Sx-\-2 


— 3a?2— 170-— 10 
— 3a?2_15a? 


2a?— 10 
2a?— 10 

0 
Hence         -         -         a?^— 3a?  +  2=0. 
Reducing    -         -         x  r=;2  or  1. 
The  three  roots  of  the  given  equation,  then,  are  5,  2,  and  1. 

2.  Reduce  the  equation  x^ — 8a:^+4a?+48=0. 

Let  x=4f .  1,  or  4  .  2. 
Substituting  successively  these  values  for  x  in  the  equa- 
tion, we  have, 
1st,   (4  .  1)^-8  .  (4  .  1)^+4  .  (4  .  l)+48:=-l .  159  ) 
2d,    (4.2)'— 8.(4.2)2+4.(4.1)  +  48=— 2.282i 

Difference  of  errors         -         -         -        — 1.123 

Then       -         -       — 1 .  1 :  0  .  1  : :  — 1  .  1  :  0  .  1.   ^ 

Hence     -         -  a?=4  .  1 — 0.1=4. 

This  value  of  a?  satisfies  the  conditions  of  the  equation  j  for, 
4'— 8.42+4.4+48=0. 
Therefore,  one  of  the  roots  is  4. 

Dividing  the  first  member  of  the  given  equation  by  x — 4, 
the  quotient  is 

a?2_4a?— 12=0. 

Reducing        -         a?=6,  or— 2. 

The  roots  of  the  equation  are  — 2,  4,  and  6 

3.  Reduce  the  equation  x^~\~16x'^-^65x — 50=0. 

^ns.  1,  5,  and  10. 

4.  Reduce  the  equation  2a:^— 16a?='+40a?2— 30a7=— 1. 

^ns.  a?  =1.2847. 

5.  Reduce  the  equation  a:5+2a?^+3a?'+4a?'+5a?=5  .  4321. 

Jlns.  x=S  .414455. 


SECT.  XII.]     RESOLUTION    OP    HIGHER    EQUATIONS.  351 

young's  METHOD  OF  RESOLVING  HIGHER  EQUATIONS. 

^S4f.  The  method  of  solving  cubic  equations  in  Art.  4-72 
is  obviously  adapted  to  equations  of  any  higher  degree  j 
and,  by  carefully  inspecting  the  properties  of  equations,  and 
the  mode  of  reduction  there  employed,  we  shall  be  able  to 
deduce,  for  the  reduction  of  equations  of  the  ;ith  degree,  the 
following  general 

RULE. 

1.  Arrange  the  coefficients  of  the  given  equation  in  a  row^ 
commencing  with  that  of  the  first  term  then  find  by  trial  the  first 
figure  of  the  root. 

2.  Jldd  the  product  of  the  first  root  figure  and  the  first  coeffi- 
cient to  the  second  coefficient ;  the  product  of  this  sum  and  the 
same  figure  to  the  third  coefficient^  and  so  on  to  the  last  coeffi- 
cient^ and  the  last  sum  will  be  the  divisor.  Multiply  this  by  the 
first  figure  of  the  root^  and  subtract  the  product  from  the  tei'm 
constituting  the  right-hand  member  of  the  equation  ;  the  remain- 
der  will  form  the  first  dividend. 

3.  Repeat  this  process  with  the  first  coefficient  and  these  sums, 
and  the  number  under  the  last  sum  will  be  the  .trial  divisor /or 
the  next  figure. 

4.  Perform  a  similar  process  with  the  first  coefficient  and  these 
second  sums,  stopping  under  the  n — Ith  coefficient,  jjgain,  per- 
form  a  similar  process  with  the  same  first  coefficient  and  these 
last  sums,  stopping  under  the  n — 2/A  coefficient,  and  so  on  till 
the  last  sum  falls  under  the  second  coefficient. 

5.  Find  now,  from  the  trial  divisor  and  the  first  dividend^  the 
next  figure  of  the  root,  and  proceed  with  the  last  set  of  sums  and 
this  new  figure  exactly  the  same  as  with  the  original  coefficients 
and  the  first  figure  in  finding  the  preceding  divisor,  and  the  sec- 
ond divisor  will  be  obtained.  Then  proceed,  as  before,  to  find 
the  SECOND  DIVIDEND,  and  so  on  till  the  work  has  been  carried  to 
a  sufficient  degree  of  exactness, 

Kote. — The  work  may  be  contracted  by  cutting  ofl'  deci- 
mals as  before 


352  ELEMENTS    OF  ALGEBRA.  [sECT.  XII. 

EXAMPLES. 

1.  Eeduce  the  equation  a;'— 3x2+7507=10000. 
Operation. 


•    0 
9 

—3 

81 

78 
162 

240 
243 

483 
29  .  44 

75 

702 

777 
2160 

10000 
6993 

|9  .  8860027,  &c.,  =x. 

9 

9 

3007 

2677 

.5616 

18 
9 

2937 
409  .  952 

329 
306 

.4384 
.1662 

27 
9 

3346  .  952 
434  .  016 

23 
23, 

.  2722 
.2616 

36.8 

.8 

512.44 
30.08 

3780  .  968 
46.  110 

106 

78 

37.6 

.8 

542  .  52 
30.72 

3827 .  07|8 
46  .  36 

28 
27 

38.4 

.8 

573  .  24 
3.  14 

3873  .  44 
3.50 

1 

3|9  . |2 

576  ..3.8  3876.  9|4 
3.1            3.5 

579  .  |5 
3 

3880  .  4 

518|3 

Kote.  —  By  bringing  down  one  period  of  decimals,  we 
have  found  the  root  to  eight  places  of  figures.  If  another 
period,  or  eight  decimals,  had  been  brought  down,  the  root 
might  have  been  found  to  twelve  plav:jes  of  figures,  or  x^=: 
9  .  88600270094. 

2.  Eeduce  the  equation  a?'+6x4— 10a?'— 112ac2—207a?=  110. 


SECT.  XII.]  RESOLUTION    OF    HIGHER    EQUATIONS.  353 

Operation, 


6 

4 

—10 
40 

30 
56 

"86 
72 

158 

88 

246 
10.56 

—112 
120 

,624 

—207 
32 

110  14.46410161. 
-700 

10 
4 

8 
344 

—175 
1408 

810 

667 . 05984 

14 

4 

352 
632 

1233 
434 .  6496 

142.94016 
133.46395 

18 
4 

984 
102. 

1667.6496 
477.4144 

9.47621 
9 . 24089 

22 

4 

1086. 
106. 

.624 
.912 

2145.0640 
79 . 3352 

23532 
23158 

26.4 
.4 

256 .  56 
10.72 

1193. 
111. 

536 

264 

2224 .  399,2 
80 .  389 

374 
232 

26.8    267.28     1304.800     2304.788  142 

4       10.88         17.453  5.434  139 


27.2     278.16     1322.2513    2310.2212 
4       11.04         17.56  5.44 


27.6     289.20     1339. 8|1       2315.66 
4         1.68         17.6  14 


218. |0    290 .^8j8    1357.4         2315. 8|0 
1.7  1.2  1 


292.16      1358.16       2131 1|5. 9 
1  1 


2|9]4         1|3;5|9 

3.  Reduce  the  equation  ar'-h2j:*+3a:'+4j:*+5a?=54321. 

Ans.  x=8  .  41445475,  &c. 

4.  Reduce  the  equation  x«-|-2x5+3x*+4ar'-|-5x--f  6x= 
654321.  ^ns.  Xz=%  .  95697957,  &c. 

5.  Reduce  the  equation  a:'— 3x<'— 2  .  5x'-|- lOx^+ar*— 9x»-j- 
2j=2.  Jlns.  x=1 ,  62599736,  &c. 

6.  Reduce  the  equation  a:«-f  10x'-h21x— 55x'— 100J?*-f- 
525x'^-8(>4x«— 630x=216.     Ans,  x=  .  79128785,  &c. 


NOTES. 


Note  A,  page  13,  Art.  2. 

The  term  Quantity  seems  to  be  used  by  writers  on  Mathematics  with 
a  great  degree  of  vagueness,  and  the  definitions  of  it  are  liable  to  many 
objections.  For  instance  :  '•  Quantity  is  a  general  term,  embracing  every- 
thing which  admits  of  increase  or  diminution."*  Now,  it  is  with  perfect 
consistency  that  the  natural  philosopher  speaks  of  "  increasing  or  dimin- 
ishing" heat :  so  mental  power  or  energy  may  be  increased  or  diminished ; 
and  so,  also,  passion,  resentment,  anger,  benevolence,  or  love  may  be  in- 
creased or  diminished.  Hence,  by  the  above  definition,  they  are  included 
under  the  term  Quantity,  and  are,  consequently,  objects  of  mathematical 
investigation.  The  incorrectness  of  this  definition  needs  no  farther  illus- 
tration. With  regard  to  Number,  we  believe  it  cannot  properly  be  included 
under  Quantity.  Dugald  Stewartf  remarks :  '•  As  to  number  and  propor- 
tion, it  might  be  easily  shown  that  neither  of  them  fall  under  the  definition 
of  Quantity,  in  any  sense  of  that  word."  Believing  the  term  Quantity  in- 
correctly applied  in  most  treatises  on  Algebra,  we  have  endeavoured  to 
substitute  the  word  number  in  its  place. 

Dr.  Reidt  suggests  a  distinction  of  Quantitj  into  proper  and  improper. 
Proper  Quantity  is  that  which  is  measured  by  its  own  kind,  such  as  ex- 
tension and- duration.  Improper  Quantity  is  that  which  cannot  be  meas- 
ured by  its  own  kind,  but  to  which  we  assign  a  measure  in  some  proper 
quantity  that  is  related  to  it.  Velocity,  density,  elasticity,  <Scc.,  may  be 
considered  as  examples  of  this  kind  of  quantity. 

Note  B,  page  16,  Art.  20. 

The  origin  of  Algebra,  like  that  of  other  sciences  of  ancient  date  and 
gradual  progress,  is  not  easily  jeceriained.  We,  however,  have  derived 
it  from  the  Arabians,  among  whom  it  was  cultivated  at  a  very  early  pe- 
riod. The  most  ancient  treatise  on  Algebra  now  extant  is  that  of  Dio- 
phantus,  a  Greek  author  of  Alexandria,  who  flourished  about  A.D.  360, 
and  wrote  thirteen  books,  six  of  which  are  now  extant.  The  following 
is  a  list  of  8ome  of  the  early  writers  on  Algebra  :  Pisanus,  1400 ;  Lucas 
de  Burgo,  1476  ;  Scipio  Ferreus,  1505  ;  Nicholas  Tartalca,  1539  ;  Cardan, 
1545;  Xylander,  1575;  Bachet,  1621  ;  M.  Fermat,  1670.  Of  later  date, 
writers  have  been  abundant ;  and  among  them  may  be  ranked  some  of 

*  Davies'  Boardon.  t  Works,  vol.  ii.,  p.  364.  i  Eway  on  Quautily. 


356  NOTES. 

the  most  distinguished  mathematicians  and  philosophers,  such  as  New- 
ton, Euler,  Des  Cartes,  and  a  host  of  others. 

Note  C,  j)age  136,  Art.  244. 
The  theory  of  indetermination  is  productive  of  several  important  con- 
sequences. 

The  remark  here  made  upon  indetermination  wrill  aid  the  student  in  his 
analysis  of  several  curious  problems.  For  instance,  we  might  cite  the 
process  by  which  two  is  made  to  appear  to  be  equal  to  one  ;  thus, 

Let a=:l,  and  x=\. 

Then a:=a; 

Multiplying  by  a:         .         _         .         -     x^=^ax; 
Subtracting  a^  from  both  members,  x^ — a^=.ax — a^ ; 
Resolving  into  factors        -     (a:+a)  [x — a)^=a[x — a) ; 
Dividing  by  x — a       -        .        -  x-\-a^=a ; 

Restoring  values  of  a:  and  a       -        -  1  +  1^1,  or  2=1. 
The  fallacy  in  the  above  reasoning  will  be  easily  detected ;  for,  express- 
ing the  division,  we  have 

a{x—a) 
X — a 
which,  since  x=a,  becomes 

.       aXO    0 

The  above  case  is  not  singular.     Take  the  identical  equation 

10=10 : 
Resolving  into  terms       -        -        -     8+2=8+2 ; 

Transposing 2—2=8—8; 

Resolving  2d  member  into  factors         2 — 2=4(2 — 2) ; 
Dividing  by  2—2 1=4. 

Note  D,  page  257,  Art.  391. 

The  invention  of  Logarithms  is  undoubtedly  due  to  John  Napier,  Baron 
of  Merchiston,  in  Scotland,  who  gave  it  to  the  world  in  a  book  written  in 
Latin,  and  entitled,  "  Mirijici  Logarithmorum  Canonis  Descriptio,  ejusque 
usus  in  utraque  Trigonomctria,  ut  etiam  in  omni  Logistica  Mathematica, 
Amplissimi,  Facilimi,  et  Expeditissimi  Explicatio.  Auctore  ac  inventore 
Joanne  Nepero,  Baronne  Merchislonii. 

Note  E,  page  262,  Art.  399. 

Formula  :  logarithm  (n+l)=log.  n+2  M  |  ^-^-^+-^^-^^3-+^^^- 

1  1 1_ > 

'^7(2«+iy^9(2'/j+l)»'^ll(2/i+iy"'^^-  } 


NOTES.  367 

The  reductioQ  of  the  above  formula  was  not  introdaced  into  the  body 
of  the  work,  on  account  of  its  length  and  complexity. 

Let  a  represent  the  base  of  the  system,  t+1  be  any  number  in  the  com- 
mon arithmetical  scale,  and  x  its  logarithm ;  then  (Art.  )  a'=r+l. 
Again,  let  a=14-6,  then  (l+6)'=l+r;  and  to  find  the  log.  of  l+u,  we 
must  solve  this  equation,  in  which  x  is  the  unknown  quantity. 

Involving  both  members  to  the  power  m,  we  have 

(I4.i)"«=(l+t,)"'; 

mrfmx— 1)     ,,    mx(mT— l)(mr— 2) 
Expanding,      l+7iu:i+— ^-^ *X4»+ ^ 'xA»+,&c. 

fn(m— 1)      .    m(m— l)(m— 2)      _     , 
=l-t-mp+    ^  g    ^Xi>'-H— ^ ^Xr'+,  <Sw5. 

Rejecting  1  from  each  member,  and  dividing  by  m, 

or      -      x(H.(^W^-^^=^*'+.*c.,)=H-^^-+ 

("-'f-^W.^c. 
Now,  let  m=0,  and  we  shall  have 

Dividmg         -        -    ^=4_^4,^j^3_^44^^^s— 4,c— 
But  a:=log.  (r-|-l) ;  therefore, 

log.  ^»-t-ij— ^_^^.^j^,_^j«_^l^»_^  ^c. 

Or,  since  a=64-l, 

Incr   r.  I  n     ^-^'-|-^t>»-^t.*-f  j^r'-^,  &C. 

Or,  if  we  take  M=, r^— -, — L_ t^ r— 

(a_l)-i(a-l)»4-i(a-l)»-,&c., 

log.  (r-f  l)=M(r-ir»4-Jr'-U*4-»p»— ,  &c.)  (A.) 

Since  a  is  a  constant  quantity,  M,  which  is  termed  the  modulus  of  sys- 
tem, must  also  be  a  constant  quantity. 

^=2-;302k509=*^294482. 

Or,  more  correctly, 

M— 434294481903251827651 1289189166. 

This  is  the  series  (A)  earliest  known  for  the  calculation  of  logarithms 
But  the  difficulty  with  it  is  that  it  will  either  diverge,  or  not  converge  so 
quickly  as  to  make  the  summation  of  a  few  terms  of  it  a  irafficient  ap- 


358  NOTES. 

proximation  to  the  value  of  x  or  log.  {v-\-l),  unless  tj  be  a  proper  fraction 
sufficiently  small.  If  v  be  nearly  equal  to  1,  the  series  converges  too 
slowly  to  be  of  any  use ;  and  if  v  be  greater  than  1,  the  series  diverges, 
and  is,  consequently,  useless. 

We  may,  however,  transform  this  series  (A)  into  others,  so  as  to  ob- 
tain a  series  that  will  apply  in  every  possible  case.  For  if  1 — v  instead 
of  l-\-v  be  used,  we  shall  obtain 

■»=    v^ 
log.  {l—v)=M{—v—-—-—\v*—^v'^—,  &c.) 


But  log.  (1 — v)=: — log.- —  ;  therefore 


=—M{v+^v''-\-lv^+\v*-\-^v^+,  &c.) 

1 


log.— -=M(?j+ii;='-}-^2)3-|-iv*-|-|«5+,  &c.)  (B.) 

By  adding  together  the  formulas  (A)  and  (B),  and  observing  that  log. 
{li-v)+  log.  j^=  log.  YZT^,  we  have 

log.  J-^=M(2r+iv3+fr*+|r^+,  &c.) 


=2M[v-hiv'+^v'+\v-'+,  &c.)  (C. ) 

Again,  let  us  put  — =-,  then  «=-—  ,  and  substituting  in  the  above 
formula,  it  becomes 

Transposing      -      log.  u^log.  ^+2M  ^^^H-j  ^^-^y +|  ^^-=^y^ 

Now,  letting  u=n-\-l  and  t—n,  we  shall  have  u—t=l,  and  u-\-t=2n-{-l ; 
substituting  these  values  in  the  preceding  formula,  it  becomes 

log.  (n+l)=log.  «+2M  ^^_j^g-^^g^^+^.jj,+,  &c.^ 

This  series  evidently  converges  very  rapidly,  even  when  w=l ;  but 
converges  more  rapidly  as  n  increases.  Hence,  having  found  the  log- 
arithm of  any  number,  we  may  easily  find  the  logarithm  of  the  next 
higher  in  the  natural  series  of  numbers  by  the  application  of  this  formula. 
See  Edinburgh  and  Recs's  Encyclopedias. 


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